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EXPERIMENTAL TESTS OF MATHEMATICAL 
ABILITY AND THEIR PROGNOSTIC VALUE 



BY 

AGNES LOW ROGERS 

M.A. (St. Andrews), Moral Sciences Tripos (Cambridge) 

Ph.D. (Columbia) 

Lecturer in Educational Psychology, Teachers College 



TEACHERS COLLEGE, COLUMBIA UNIVERSITY 
CONTRIBUTIONS TO EDUCATION, No. 89 



PUBLISHED BY 

S^rarlfrrs (BalW^t, Ql0lttmfatH InitirrBtty 

NEW YORK CITY 

1918 



T7f4 



Copyright, 1918, by Agnes Low Rogers 



APR 18 1918 



OCf.A492975 



ACKNOWLEDGMENTS 

It would be difficult to acknowledge all that I owe to others in 
this study, but to Principal Stuart H. Rowe of Wadleigh High 
School and to Principal Henry Carr Pearson of the Horace Mann 
School my thanks are specially due for the permission to use the 
time of the pupils in making this investigation. To them and to 
the teachers of these schools I am grateful for their cooperative 
aid, which ensured satisfactory conditions for testing. 

I desire also to record my thanks to Professor E. L. Thorndike 
for helpful supervision throughout the conduct of this research. 
To Professor H. A. Ruger I am likewise indebted for friendly 
assistance in many phases of the study. 



ni 



CONTENTS 

I. Summary of Previous Work 1 

II. General Conditions of the Present Investigation, Appli- 
cation OF THE Tests and System of Scoring .... 15 

III. The Analysis of Mathematical Ability 42 

Reliability of the tests 

Method of calculating coefficients of correlation 
Correction of coefficients of correlation for attenuation 
Development of composite measure of mathematical 

ability 
Relative value of each test as a measure of mathematical 

ability 
Determination of the characteristics in a test which produce 

high correlation with mathematical ability 
Correlation of the composite for mathematical ability 

with school marks 

IV. The Prognosis of Mathematical Ability 90 

V. Summarized Conclusions 96 

Appendix 97 

The original scores 
Instructions to subjects 

Practical use of the sextet of tests for diagnosing mathe- 
matical ability 



TESTS OF MATHEMATICAL ABILITY AND 
THEIR PROGNOSTIC VALUE 

CHAPTER I 
SUMMARY OF PREVIOUS WORK 

The decision of every question of moment in education 
depends upon both psychological and sociological considerations. 
As regards the latter, investigation is greatly complicated at the 
present time by the far-reaching character of the changes that are 
taking place in our industrial and social life. So rapid and so 
complex are these changes that recommendations based upon yes- 
terday's situation may prove ill-adapted to that of to-day. In 
this respect the psychologist has a considerable advantage over the 
sociologist, where educational guidance is concerned; for, how- 
ever variable and elusive it may be, the original nature of man is a 
more stable thing than the environment to which it is exposed. 
The task of discovering what can be known of the innate abili- 
ties of the individual presents fewer obstacles to the scientific 
investigator and once attained it will remain a permanent pos- 
session and unfailing fingerpost for the educator. 

In no sphere is this knowledge more desirable and necessary at 
the present time than in the high school subjects and particularly 
in mathematics. Reforms of a far-reaching character are already 
planned or in process and it is important that such psychological 
considerations as bear upon a satisfactory scheme should be 
ascertained. In our present comparative ignorance of the abili- 
ties involved in mathematical work, we lack one important means ^ 
of estimating the significance of the reconstructions proposed in 
this field. 

This study is a partial contribution towards supplying this 
need. Its purpose is to make an analysis of the abilities involved 



2 Tests of Mathematical Ability and Their Prognostic Value 

in high school mathematics, to determine their efficiency and 
status, their interrelations and also their connection with certain 
other forms of mental capacity. Primarily it is directed to dis- 
cover dynamic and quantitative relations between mathematical 
abilities rather than to show how we think in mathematics 
from the standpoint of analytic or structural psychology. It is 
to be distinguished, therefore, from the work on the thought 
processes of the Wiirzburg School, since it does not attempt to 
analyze the content of thought in mathematical thinking, while 
it seeks to determine the functional relations between mathe- 
matical abilities and their connection or lack of connection with 
certain other mental abilities. It is not concerned with the devel- 
opment of ideas of number or space in the child, which recent 
writers on the Psychology of Mathematics have considered at 
some length ^ and which has importance for the elementary school 
teacher. Neither is it an investigation of mathematical genius, 
nor even a consideration of the capacities called into play in 
higher mathematics, though there are certainly important fea- 
tures common to high school mathematics and higher mathe- 
matics. If we exclude arithmetic, which has of late received 
considerable attention, we find with few exceptions that most of 
the publications upon the psychology of the subject have treated 
of creative ability in mathematics and of the nature of the capaci- 
ties demanded by higher mathematics and further that the balance 
of opinion favors the view that there is a radical difference 
between high school mathematics and higher mathematics. For 
example, Betz ^ asserts that "School mathematics has extremely 
little to do with real mathematical thinking." (Nun hat aber die 
Schulmathematik mit dem eigentlich mathematischen Denken nur 
ailsserst wenig su tun.) Henri Poincare ^ in like vein writes, 
"Many children are incapable of becoming mathematicians, to 
whom, however, it is necessary to teach mathematics. All we can 
do is to work with them, adapting ourselves to their properties." 
By mathematician Poincare does not mean those possessing crea- 
tive genius alone; under the term he includes those capable of 

1 See Bibliography, Howell, H. B., The Pedagogy of Arithmetic, New 
York, 1914. 

2 Betz, W., Uber Korrelationen, Ztsch. fur Ang. Psych., beihef te : 1911. 
8 Poincare, H., The Foundations of Science, New York, 1913, tr. by 

Halsted, G. B. 



Summary of Previous Work 3 

understanding higher mathematics, though they cannot do orig- 
inal work in that field. William Brown similarly affirms : * 
"There is good reason for thinking that school mathematics and 
higher mathematics relate to different forms of ability and should 
be clearly distinguished from one another." These writers 
further contend, as likewise does Katz,^ in reviewing the whole 
subject, that whereas higher mathematics demands special ability, 
any intelligent child can master the mathematics required in the 
secondary school, provided he exerts himself earnestly. 

The various treatises upon mathematical genius that have ap- 
peared have utilized the methods of observation and introspec- 
tion. They present two main strands of thought. On the one 
hand, ability in mathematics is held to be a special fundamental 
capacity, independent of other mental capacities — a view taken 
by Mobius,® for example. On the other hand, it is regarded 
merely as consisting in an "unusual ease in performing certain 
thought operations." 

A variety of opinions exists as to the nature of these funda- 
mental processes involved in mathematical thinking. According 
to Wundt ^ the essence of geometrical ability is the union of con- 
crete imagination with deductive understanding. This partic- 
ular combination produces the analyzing type of mind, character- 
istic of scientists and geometricians. The ability to synthesize 
together with inductive ability makes the discoverer, while the 
former coupled with deductive ability makes the speculative 
thinker. Mathematicians may belong to either class. 

In 1894 Professor Calkins ® communicated to the Educational 
Review a study of the Mathematical Consciousness by Wellesley 
students. The main conclusions reached were as follows : "Con- 
crete memory characterizes the mathematically inclined and be- 
longs to geometricians to a greater extent than to algebraists. 
Though imagination is the foundation of every mathematical as 
of every conscious process and though memory is at least as com- 
mon among mathematicians as among average individuals, the 

* Brown, W., The Psychology of Mathematics, Child Study, 6: 26. 
^ Katz, D,, Psychologic und mathematischer Unterricht, Leipzig, 1913. 
6 Mobius, P. J., Uber die Anlage zur Mathematik, Leipzig, 1900. 
■^ Wundt, W., Grundziige der Physiologische Psychologic, III : 636. 
s Calkins, M. W., A Study of the Mathematical Consciousness, Educa- 
tional Review, VIII : 269. 



4 Tests of Mathematical Ability and Their Prognostic Value 

essential characteristic of the student of mathematics is the power 
of thought, of identification, comparison and reasoning. The 
ability to notice the similarity or dissimilarity between objects 
or relations and to classify them accordingly is prominent. In 
algebra the given problem must be classified as one to whose solu- 
tion certain rules apply. In geometry a theorem must be demon- 
strated. This classifying power is strong in mathematics. Fur- 
ther, geometry is more congenial to the true mathematician than 
algebra, and mathematics involves the possession of every sort of 
ability." 

In 1900 Mobius ^ suggested that mathematical talent is char- 
acterized by exceptional ability in understanding relations of 
number, in judging relations of size, and in concrete imagery. 

In 1905 there appeared in the French mathematical journal 
I'Enseignement Mathematique the first of a series of articles pub- 
lished at intervals until 1908 under the general title, "Enquete sur 
la Methode de Travail des Mathematiciens." These consisted 
of replies to a questionnaire summarized by H. Fehr and others. 
They contained many interesting facts about the mental habits 
and methods of work of mathematicians, but suffer from the 
psychological superficiality of such studies. 

In 1908 Henri Poincare published a suggestive discussion of 
L'Invention Mathematique in the Bulletin de VInstitut general 
Psychologique. This was the forerunner of several papers upon 
the nature of mathematical ability. In these Poincare ^° con- 
tends that "it is impossible to study the works of the great mathe- 
maticians or even those of the lesser, without noticing and dis- 
tinguishing two opposite tendencies, or rather two entirely dif- 
ferent kinds of minds. The one sort are above all preoccupied 
with Logic. . . . The other sort are guided by intuition. . . . The 
method is not imposed by the matter treated. Though one often 
says of the first that they are analysts and calls the other 
geometers, that does not prevent the one sort remaining analysts 
even when they work at geometry, while the others are still 
geometers, even when they occupy themselves with pure analysis. 
It is the very nature of their mind, which makes them logicians 
or intuitionalists, and they cannot lay it aside when they approach 

» Op. cit. 
10 Op. cit. 



Summary of Previous Work 5 

a new subject. . . . Nor is it education which has developed in 
them one of the two tendencies and stifled the other. The mathe- 
matician is born, not made, and it seems he is born to be a 
geometer or analyst. . . . Among our students we notice the same 
differences : some prefer to test their problems by analysis, others 
by geometry. The first are incapable of 'seeing in space,' the 
others are quickly tired of long calculations and become per- 
plexed." 

Poincare also maintains that mathematical ability is not due 
merely to a very sure memory nor to a prodigious power of at- 
tention. If it were, every mathematician would also be a good 
chess player and likewise a good computer and this is far from 
being the case. 'Tn a word my memory is not bad," he writes, 
"but insufficient to make me a good chess player. Why does it 
not fail me then in a difficult piece of mathematical reasoning, 
where most chess players would lose themselves? Evidently be- 
cause it is guided by the general march of the reasoning. A 
mathematical demonstration is not a simple juxtaposition of 
syllogisms, it is syllogisms placed in a certain order, and the 
order in which these elements are placed is much more important 
than the elements themselves. If I have the feeling, the intuition, 
so to speak, of this order, so as to perceive it at a glance, the 
reasoning as a whole, I need no longer fear lest I forget one of 
the elements, for each of them will take its allotted place in the 
array and that without any effort of memory on my part." Ac- 
cording to Poincare, it is this intuition of mathematical order 
which distinguishes the mathematician from other men. 

Hiither,^^ writing in 1910, asserts that it is extraordinary de- 
velopment of concrete imagery, synthetic imagination and mathe- 
matical understanding that marks the mathematician, while in 
1911 Betz presents the theory that the mathematical type of 
mind is characterized by a special clearness of certain minimal or 
highly abstract ideas and by the capacity to vary these with pre- 
cision and to manipulate them with facility. "As soon as it gets 
to be a matter of discovering mathematical principles independ- 
ently," he says, "then stands the unmathematical before insuper- 

^^ Hiither, A., Uber das Problem einer psychologischen und padagogis- 
chen Theorie der intellektuellen Begabung, Archiv. fiir die gesamte Psy- 
chologies XVIII : 193. 



6 Tests of Mathematical Ability and Their Prognostic Value 

able barriers, then he is simply incapable, and even a person of 
average ability as regards mathematics comes sooner or later upon 
a problem, which he cannot grasp without external aid and which 
a better mathematician can solve with relatively little effort. 
The mental state has a certain resemblance to the situation where 
one tries to hold fast in a visual image certain details, of which 
straightway not a significant trace is visible; but in the case of 
mathematical thinking it is not a matter of visual memory images, 
but of peculiar ideas, which are felt rather than seen and which 
I, in another connection, have called Minimal Ideas." 

Akin to the foregoing studies is Judd's ^^ treatment of the 
psychology of mathematics, inasmuch as it presents a survey based 
in part upon experimental work of the psychological processes 
underlying mathematics. It describes typical mental reactions 
involved in mathematical thinking, and analyzes the psychological 
implications of the text-books in use and of current class-room 
procedure. Judd asserts that the abilities demanded by algebra, 
geometry, and arithmetic are essentially different in character, 
each representing some forms of mental activity not included in 
the others. 

An entirely dissimilar method of approach has been made by 
those workers who have been interested in the establishment of 
standards and scales, as likewise by those who have resorted to 
the use of the objective statistical method of correlations. 

If we consider arithmetic, it appears that the work in this field 
has been extensive and has significance for mathematics in general. 
The most important results of the studies by Thorndike,^^ Stone,^* 
Bonser,^^ Courtis,^® Winch,^^ Starch,^^ and Woody ^^ are the 
demonstration of the wide range of individual differences in 
capacity and the specialization and independence of the different 
abilities involved in arithmetic. A high degree of excellence in 
the fundamental processes (addition, subtraction, multiplication, 
and division) has been shown to be consistent with a low degree 

12 Judd, C A., The Psychology of High School Subjects, New York, 
1915. 

13 See Bibliography in H. B. Howell's A Foundational Study in the Peda- 
gogy of Arithmetic, New York, 1914. 

^^Ibid. ^^Ibid. ^^ Ibid. ^Ubid. ^^ Ibid. 

1* Woody, C, Measurements of Some Achievements in Arithmetic, 
Teachers College, Columbia University Contributions to Education, No. 80. 



Summary of Previous Work 7 

of skill in arithmetical reasoning and vice versa. Indeed a similar 
variability was found to prevail among the fundamental processes 
themselves. These results led Fox and Thorndike ^^ to prophesy- 
that the abilities tested — addition, multiplication, fractions, ra- 
tional computation and problems — ^bear little resemblance to those 
of the mathematician. 

Bonser ^^ found similar results in investigating the reasoning 
ability of children in the fourth, fifth, and sixth school grades. 
Among others he gave certain tests of mathematical judgment. 
These were problems in arithmetic, stated in unusual form. He 
obtained the following correlations : 

Arithmetic Tests and Completion Tests 41 

Arithmetic Tests and Opposites 42 

Arithmetic Tests and Selecting Correct Reasons ZZ 

Arithmetic Tests and Selecting Best Definitions 26 

Arithmetic Tests and Literary Interpretation 26 

Arithmetic Tests and Spelling 24 

In the field of algebra and geometry, if we exclude the efforts to 
establish standards for algebra by Thorndike, Monroe,^^ Rugg ^^ 
and Clark, and Childs,^* and standards for geometry by Stock- 
ard ^^ and Carleton Bell,^^ we find that in all the experimental in- 
vestigations published, with four exceptions, the data have been 
school and college marks or class lists. Correlations between 
school marks in mathematics and in English and between the 
former and drawing were calculated by Smith.^^ He found in 

20 Fox, W. S., and Thorndike, E. L., The Relationship between the 
Different Abilities involved in the Study of Arithmetic, Columbia Con- 
tributions to Philosophy, Psychology, and Education, XI : 32. 

21 Bonser, F. G., The Reasoning Ability of Children of the Fourth, 
Fifth and Sixth School Grades, Teachers College, Columbia University 
Contributions to Education, No. 2)7. 

22 Monroe, W. S., Measurement of Certain Algebraic Abilities, School 
and Society, I: 393, and A Test of the Attainment of First Year High 
School Students in Algebra, School Review, XXIII : 159. 

23 Rugg, H. O., The Experimental Determination of Standards in First 
Year Algebra, School Review, XXIV : Z7. 

Rugg, H. O. and Clark, J. E., Standardized Tests and Their Improve- 
ment of First Year Algebra, School Review, XXV: 115 and 196. 

2* Childs, H. G., The Measurement of Achievement in Algebra, Bulle- 
tin, Extension Division, Indiana University, II : 6. 

25 Stockard, L. V. and Bell, J. Carleton, A Preliminary Study of the 
Measurement of Abilities in Geometry, Jour. Ed. Psych. VII : 567. 

26 Ihid. 

27 Columbia University Contributions to Philosophy, Psychology, and 
Education, XI, No. 2. 



V 



8 Tests of Mathematical Ability and Their Prognostic Value 

the case of mathematics and English the coefficient was .Z6 for 
boys and .43 for girls: for mathematics and drawing it was .16 
for boys and .12 for girls. Burris ^^ obtained between English 
and mathematics a correlation coefficient of .39 and between 
algebra and geometry a coefficient of .45. 

Brinckerhoff, Morris, and Thorndike ^* used the regents' exam- 
ination marks in order to avoid the influence of the pupils' looks, 
manners and attitude upon the teacher's judgment. All pupils 
considered were from the same school and had practically had 
the same training. The coefficients between mathematics and 
other secondary school subjects were positive and low. They 
were as follows (Burris's results are given in parentheses) : 

Mathematics and English 09 (.39) 

Mathematics and Science 07 (.41) 

Mathematics and History 26 i-^^) 

Mathematics and German 48 

Mathematics and Drawing 02 

Mathematics and Latin 31 (.40) 

Rietz and Shade ^^ found higher correlations with science and 
similar correlations with foreign languages. Between mathe- 
matics and science the coefficient obtained was .440 with a P. E. of 
.015 and between mathematics and languages the coefficient was 
.476 with a P. E. of .015. 

Similar results were obtained by H. O. Rugg ^^ in a more recent 
study. 

Subjects Correlated Value of r 

Mathematics and Descriptive Geometry 70 

Mathematics and Modern Languages 50 

Mathematics and EngHsh 40 

Mathematics and Shop-Practice 44 

Mathematics and Shop-Practice 38 

Mechanical Drawing and Shop-Practice 44 

A statistical study carried out in the pedagogical department of 

28 Ibid. 

29 Ibid. 

30 Rietz, H. L., and Shade, J., Correlation of Efficiency in Mathematics 
and Efficiency in other Subjects, The University of Illinois Studies, VI: 
301. 

31 Rugg, H. O., The Experimental Determination of Mental Discipline in 
School Studies, Baltimore, 1916, p. 93. 



Summary of Previous Work g 

Dartmouth College under the direction of F. C. Lewis,^^ in 1905, 
deserves mention on account of the departure made in method, 
y Instead of using school marks as data, tests were given in originals 
in geometry and in practical reasoning and the scores made in 
these were correlated. It may be doubted whether the tests were 
adequate measures of the abilities in question and the method of 
correlation was misleading. The pupils of each of twenty-four 
groups were arranged in two series, the first according to their 
ranking in mathematical reasoning, and the second according to 
their ranking in practical reasoning. It was found that of the 
first five mathematical reasoners from each group 63 per cent., 
that is, 76 persons, were at the foot of the practical reasoning 
series, conspicuous for their inefficiency in practical reasoning; 
and of the pupils at the foot of the mathematical reasoning series, 
47 per cent, were conspicuous for their positions at the head of 
the practical reasoning series. 

These results have been subjected to criticism by Rietz,^^ who 
points out that Lewis's conclusion that they furnish convincing 
evidence "that students able in mathematical reasoning are not 
even generally able in practical reasoning and law" is far from 
justified, since not only were the data relatively few, but the 
coefficients of correlation derived from them (.38 to .675) are 
both positive and significant. 

None of the preceding studies made any correction for the at- 
tenuation of the coefficients of correlation due to chance inac- 
curacies in the original measures. The true relationships between 
the mathematical abilities and the other abilities investigated are 
probably much higher than these crude coefficients indicate. We 
can judge to what extent the latter would be raised by correction, 
from the few corrected coefficients calculated by Bonser from one 
of the groups he examined. In the case of the data from the 
boys in Grade 6A two methods of correction were applied,^* and 
the following coefficients were obtained. The crude coefficients 
are also given for purposes of comparison. 

32 Lewis, F. C, A Study in Formal Discipline, School Review, XIII: 
281. 

33 Rietz, H. L., On the Correlation of the Marks of Students in Mathe- 
matics and in Law, Jour. Ed. Psych., VII : 87. 

3*Thorndike, E. L., Theory of Mental and Social Measurements, New 
York, 1913, 177. 



lo Tests of Mathematical Ability and Their Prognostic Value 

Average 
Gross Corrected Coefs. of Cor. 
Coefs. Meth.l Meth.2 Coefs. 

Arithmetic Tests and Completion Test.. .31 .55 .37 .46 

Arithmetic Tests and Oppo sites 43 1.04 .57 .81 

Arithmetic Tests and Selecting Correct 

Reasons 00 .39 .20 .10 

Arithmetic Tests and Selecting Best Def- 
initions 31 .99 .45 .72 

Arithmetic Tests and Literary Interpreta- 
tion 25 .46 .36 .41 

Arithmetic Tests and Spelling 19 .50 .19 .34 

The following coefficients of correlation obtained by Spear- 
man ^^ in an investigation into the nature of general intelligence 
are still larger. In the case of the school subjects examination 
marks were used as data. 

Crude P.E. Corrected 

Mathematics and Pitch Discrimination 39 .03 

Mathematics and Pitch Discrimination (musicians 

only) 45 .03 .61 

Mathematics and Mathematics (reliability coefs.) . . .88 .01 

Mathematics and Classics 70 .01 .81 

Mathematics and French 67 .01 .78 

Mathematics and English 64 .01 .74 

Mathematics and General hitelligence .86 

An attempt to secure a more complete and detailed analysis of 
mathematical intelligence was made in 1910 by William Brown.^® 
He used the same statistical method of correlation, obtaining his 
data from a school examination in algebra, geometry, and arith- 
metic. He corrected the papers, however, in two ways, according 
to ordinary school standards, and also according to a differential 
system of marking based upon an introspective analysis of the 
intellectual processes involved in answering. The latter method 
is perhaps even more open to criticism than the former, since 
there are obvious defects in the "psychologizing" of examination 
papers. 

The following results were obtained. 

35 Spearman, C, "General Intelligence," Objectively Determined and 
Measured, Amer. Jour. Psych. XV : 275. 

36 Brown, William, An Objective Study of Mathematical Intelligence, 
Biometrika. VII : Z67. 



Summary of Previous Work 1 1 

r P.E. 

Arithmetic and Algebra 79 ,03 

Geometry and Algebra 66 .04 

Geometry and Arithmetic 58 .05 

Memory of preceding propositions and power of applying 
them and Recognition of necessity of generality 
of proof and power to recognize general relations 

in a particular case 81 .02 

Accuracy in Arithmetic and Accuracy in Algebra 69 .04 

Memory of preceding propositions and power of applying 
them and Power to do sums in percentage and pro- 
portion 59 .05 

General memory of rules and power to apply in Arith- 
metic and General memory of rules and power to 

apply in Algebra 49 .06 

Power to do sums in percentage and proportion and 
General memory of rules and power to apply in 

Algebra 49 .06 

Recognition of necessity of generality of proof and 
power to recognize general relations in a particular 
case and Power to do sums in percentage and pro- 
portion 44 .06 

Memory of constructions in Geometry and power to do 

sums in percentage and proportion 26 .07 

Brown's principal conclusions were that geometrical and 
algebraic ability are not related, save through their connection 
with arithmetical ability, that memory of preceding propositions 
is the central ability in geometry, being related most intimately to 
other forms of geometrical ability, that the difference between 
geometrical ability and algebraic ability justifies Poincare's theory 
that mathematical reasoners fall into two distinct types, the geo- 
metrical or intuitional and the analytical or logical, and that the 
"balance of evidence seems to be in favor of the existence of a 
special capacity or faculty underlying mathematical ability, dis- 
tinct from and with no essentially close connections with other 
forms of intellectual capacity." ^^ 

In 1913 appeared a study of great practical interest, T. L. 
Kelley's ^^ "Educational Guidance." Using the method of the 

37 Brown, W., An Objective Study of Mathematical Intelligence, Bio- 
metrika, VII : 352 and The Psychology of Mathematics, Child Study, VI : 
26. 

38 Kelley, T. L., Teachers College, Columbia University Contributions 
to Education, No. 71. 



12 Tests of Mathematical Ability and Their Prognostic Value 

Regression Equation, the author showed how prognosis of abiUty 
in mathematics, EngHsh, and history could be made on a basis of 
past school record, teachers' estimates of ability, and the results 
of tests in these subjects. Of these three means of prognosis past 
school record was found to be most satisfactory, inasmuch as the 
prognoses so derived corresponded most closely with the actual 
future achievements of the individuals tested. It may, however, 
be objected, even when allowances are made for the difficulty of 
measuring abilities in a field which is new to the persons ex- 
/amined, that the particular mathematical tests used in this study 
were far from adequate measures of geometrical and algebraic 
abilities and that with better tests the relative values of school 
marks and tests as means of prognosis might be reversed. In 
general, the possible independence of abilities, which superficially 
seem closely akin, and the possible identity of abilities which 
superficially seem notably disparate, demonstrate the need for a 
many-sided gauge of mathematical ability. Existing statistical 
studies unequivocally suggest that here we should act on the 
principles of dynamic psychology, upon which Alfred Binet ^^ 
relied in measuring general intelligence. We cannot assume that 
a single test or even two or three tests can give an adequate 
measure. The only safe and sure method is to cover as many 
phases of mathematical skill and insight as possible and pool the 
results. Theorists relying upon introspection have sought after 
a single clue, a distinguishing mark, which would differentiate 
mathematical ability from all other forms of ability and constitute 
the essence of mathematical talent, but this would seem to be a 
questionable assumption and one which itself demands experi- 
mental investigation. 

This brief summary of the literature serves to show the pres- 
ent position of our knowledge in this field. It will be seen that 
it offers little more than a number of suggestions as to the nature 
of mathematical capacity and but scanty evidence as to the 
dynamic connections between mathematical abilities and other 
abilities. The results obtained by those using the methods of 
introspection and observation are for the most part speculative 

39 Binet, A., Les Idees Modernes sur les Enfants, Paris, 1911, 117 and 
242, and L'Annee Psychol. XVII: 183. 



Summary of Previous Work 13 

in character. Mobius, Poincare, Hiither, Betz and others have 
advanced certain theories as to what constitutes the essence of 
mathematical capacity, but these are only interesting hypotheses, 
which await confirmation. The objective method of correlation 
has yielded more fruitful results, but since conclusions cannot be 
more accurate and reliable than the data from which they are 
derived, the fact that school and college marks were used in the 
bulk of statistical studies greatly limits their value and significance. 
Where more accurate investigation has been attempted, notably by 
William Brown, the ideals of scientific measurement have not been 
fully attained.**' 

The experiments forming the subject matter of this study are 
directed towards furnishing an answer to some of the more im- 
mediate and pressing problems in this field. Thus the nature of 
mathematical ability demands further investigation. It is im- 
portant both from the theoretical and practical standpoint that 
the dynamic relations between mathematical abilities should be 
more accurately known. Without this information, control over 
the learning process is greatly limited. We need to determine 
whether there is one outstanding ability, which is the funda- 
mental capacity in mathematical work. We require to know 
whether mathematical talent is such that it can function with ap- 
proximately equal facility in relation to all kinds of material or 
w^hether it is in its very nature specialized, and tied down to 
definite objects and situations. We also stand in need of some 
means of making prognoses of mathematical efficiency or insuf- 
ficiency with a view to educational guidance. 

Thus the scope of this study is comprehended by a considera- 
tion of the following questions : 

1. The development of tests which are reliable measures of the 

principal forms of mathematical activity required in high 
schools. 

2. The kind and amount of correlation of these forms of activity 

with one another. 

3. The relative value of each test as a measure of mathematical 

ability. 

^0 See Thorndike, E. L., Theory of Mental and Social Measurements, 
New York, 1913, Chap. IL 



14 Tests of Mathematical Ability and Their Prognostic Value 

4. The determination of the characteristics in a test which make 

for high correlation with mathematical ability. 

5. The selection of a group of tests which will give a sufficiently 

accurate prognosis of mathematical ability. 



CHAPTER II 

GENERAL CONDITIONS OF THE PRESENT INVESTI- 
GATION, APPLICATION OF THE TESTS 
AND SYSTEM OF SCORING* 

The Subjects 
The subjects who were examined in the present investigation 
comprised : 

(1) A group of fifty-three girls attending the Wadleigh High 
School. Their ages ranged from twelve and a half years to six- 
teen years and eight months, the average age being fourteen and 
a half years. They had had five months' training in formal al- 
gebra, but no geometry. The first application of the tests was 
made in the third week in June, 1916. As a rule the tests were 
given in the regular mathematics hour or in a study period, 
save in the case of certain tests of language ability, which were 
administered during the English class time. In general, the 
groups tested numbered twenty-five to thirty and the duration 
of examination was thirty-five minutes. All the mathematics tests 
were given by the writer, as also were the tests of verbal ability 
with the exception of the Thorndike Reading Scale Alpha 2 and 
the Trabue Language Completion Scales L and M, which have 
been so carefully standardized in method of application that the 
difference in results due to different experimenters is negligible.^ 

The second application of the tests was made in October, 1916. 
The summer vacation had intervened and had been considerably 
extended owing to the epidemic of infantile paralysis in New 
York City. Little further training in mathematics, therefore, 

* In the making of the tests helpful constructive suggestions and criti- 
cism were given by Miss Livia Ferrin. 

1 The writer's thanks are due to Dr. Lorle Stecher, who administered 
the tests of verbal ability and to Miss Helen D. Romer, who rendered 
helpful assistance in the distribution and collection of the tests. 

15 



1 6 Tests of Mathematical Ability and Their Prognostic Value 

had been received. In consequence of the late reopening it was 
necessary to make the second appHcation of the tests after school 
hours. A small sum of money was offered to the subjects to in- 
duce them to stay voluntarily. Two and a half hours' testing 
on two afternoons from 2 :30 to 5 p. m. completed their examina- 
tion. On these occasions precautions were taken to avoid fatigue 
by conducting short breathing exercises at intervals of forty 
minutes, the usual duration of a class period. The interest of 
the group was remarkable. The girls entered into the work with 
zeal and earnestness. The difference in the conditions of the 
two applications will have to be remembered, however, when we 
come to consider the results and compare them with those ob- 
tained from the second group tested. In all probability they 
effected a reduction in the coefficients of correlation derived. 

(2) A group of sixty-one pupils in the Horace Mann High 
School for Girls. They ranged in age from twelve years and ten 
months to sixteen years and eleven months, the average age being 
fourteen and a half years. Two-thirds of the group had had 
five months of intuitional geometry and five months of algebra. 
As in the case of the Wadleigh High School group, each test 
was given in duplicate, the second application generally following 
twenty-four hours after the first and never more than a week 
later. The tests were given either in the regular mathematics 
hour, when approximately twenty-five girls were examined to- 
gether, or in a study period, in which the group as a whole partici- 
pated. For this, as for the former group, all the tests were ad- 
ministered by the writer with the exception of the Thorndike 
Reading Tests, which were given by the English teacher as an 
English class exercise. In the case of both groups the scores 
obtained were made known to the pupils and considerable inter- 
est was shown in these. 

Tests With Their Administration and Scoring 

The tests used in this study were selected or devised to touch 
as many forms of mathematical achievement as possible in the 
particular groups examined. They can be divided into three 
chief classes. Six are tests of algebraic abilities and with these 
may be grouped a test of skill in problems in arithmetic and a test 
of ability to reason with symbols. The latter involves the selec- 



General Conditions of the Present Investigation 17 

tion of relevant data in order to deduce the required conclusion 
and is thus akin to the type of reasoning which predominates in 
algebra. The second class consists of five tests of geometrical 
abilities, three of which measure intuitive grasp of spatial rela- 
tions, one the ability to infer with spatial data and one the power 
to generalize from spatial facts. 

Several of the tests, it will be seen, resemble ordinary class- 
room exercises, save that they are arranged in an order of in- 
creasing difficulty and were applied under controlled conditions. 
The other mathematical tests were designed to measure abilities 
which obviously play a part in higher mathematics or which pre- 
vious psychological investigators have stated to be essential fac- 
tors in mathematical ability. 

In the case of each new test, prolonged preliminary trials ^ 
were made and as a result some of the tests were discarded as 
unreliable or impracticable. Those were retained which gave 
the clearest indication of being adequate measures of abilities im- 
portant in the mental equipment of the student of mathematics. 
Eventually the following tests were adopted. 

1. Algebraic Computation 7. Geometry 

2. Matching Equations and Prob- 8. Superposition 

lems 9. Symmetry 

3. Matching Nth Terms and Series 10. Matching Solids and Surfaces 

4. Interpolation 11. Geometrical Definitions 

5. Missing Steps in Series 12. Arithmetic Problems 

6. Inference with Symbols 13. Reasoning 

In addition to these tests of mathematical activities a third 
series of tests of language ability was given. The purpose under- 
lying their application was to discover how far weakness in mathe- 
matics depends upon or is connected with inferiority in command 
of the vernacular. The tests of language ability used were the 
following : 

1. Mixed Relations 3. Trabue Language Scales 

2. Logical Opposites 4. Thorndike Reading Tests 

The coefficients of reliability obtained from correlating the 

2 The writer is indebted to Principal J. Cayce Morrison for the oppor- 
tunity to make these preliminary trials of the tests in Chatham High 
School. 



1 8 Tests of Mathematical Ability and Their Prognostic Value 

two applications of the same test in the case of the Wadleigh 
High School group were in several instances too low to be satis- 
factory. This was accounted for in part by the differences in 
the conditions of application, but was also attributed to remediable 
defects in the tests, such as their short duration. They were 
therefore extended not only in time, but also in difficulty, before 
being applied to the second group. 

A brief description of the tests applied follows. 

Algebraic Computation Test: 

The following test was constructed for the purpose of measur- 
ing efficiency in algebraic computation. After each problem are 
given directions for grading it. 

ALGEBRAIC COMPUTATION TEST 
(1) 

1. Let C stand for the cost and SP for the selling price and G for the 
gain. Then G=zSP—C. 

What is G if C is $10 and SP is $12.50? 

Ans (Score 1 or 0) 

2. Let L stand for the length of a room and W stand for the width 
and SF for the area. Then SF=LXW. 

What is SF when L=18 feet and ^=10 feet? 
Ans (Score 1 or 0) 

3. If a=2 and 6=3 and c=5 and d=l, write the values of: 

5a Ans (Score 1 or 0) 

2a— d Ans (Score 1 or 0) 

a-\-h-{-d 

T— ^w-y (Score 1 or 0) 

a 

2a+c 

— zj — Ans (Score 1 or 0) 

2c Ab 

— - — Ans (Score 1 or 0) 

a 3d 

4. 2^ + 5^ — 3^ + 9^ — 2^ = how many ^? 

Ans (Score 1 or 0) 

5. 3a-\-4a+7a—5a-\-6a=how many o's? 

Ans (Score 1 or 0) 

6. 3x+3y-\-4s+22+2x=how many ^'s, how many y's and how many ^'s? 

Ans (Score 3, 2, 1, or 0) 



General Conditions of the Present Investigation 19 

7. Zx-\-2y — Zz-{-7z—2x-\-9y-\-Zx=:ho-w many x's, how many :y's and how 

many ^'s? 

Ans (Score 3, 2, 1, or 0) 

8. If 6^=30, what does ^r=? Ans (Score 1 or 0) 

9. If ;r— 2=5, what does ^=? Ans (Score 1 or 0) 

10. If 2;t:+3=15, what does ;ir=? Ans (Score 1 or 0) 

11. If 15— ;r=9, what does x^"^ Ans (Score 1 or 0) 

12. If x=4 and 2x=zS-\-y, what does 3)=? Ans (Score 1 or 0) 

13. A man is now 40 years old, how old was he 8 years ago? 

Ans (Score 1 or 0) 

14. Y stands for the number of years in a man's age now, how old 

was he 5 years ago? Ans (Score 1 or 0) 

15. If L stands for the length of a room, what is the length of a room 

4 feet longer? Ans (Score 1 or 0) 

16. If 1 pencil costs 5 cents, how many cents will B pencils cost? 

Ans (Score 1 or 0) 



ALGEBRAIC COMPUTATION TEST 
(la) 
Multiply and remove parenthesis: 

1. 4(3jr— 4)= 2. —S{—Ax-6y)= 3. -^(j:— 2)= 

Ans Ans Ans 

(Score 2, 1, or 0) (Score 2, 1, or 0) (Score 2, 1, or 0) 

Change all terms containing x to the left side of the equation and all 
others to the right side. 

1. —17;r— 12=192— 7.r+32 2. 17;ir— 38+7=33jir— 8^— 91 

Ans Ans 

(Score 2, 1, or 0) (Score 2, 1, or 0) 



Clear the following equations of fractions. Do not collect or trans- 
pose terms. 

1. —7^—2 x+\ 2. x—Z 5jr+4-l 

6 8 9 12 



Ans Ans 

(Score 2, 1, or 0) (Score 3, 2, 1, or 0) 



20 Tests of Mathematical Ability and Their Prognostic Value 

Solve the following problems for x. 

1. —Zx—2 x+2 2. 2{—4x+7) 3(3^—2) 

= Ax =3+ 

4 6 8 5 

Ans (Score 3, 2, 1, or 0) Ans (Score 2, 1, or 0) 

Find the values of both unknowns in the following equations : 

1. 7;^__43,=:12 
%x—Sy= 

Ans. : ;r= y= (Score 3, 2, 1, or 0) 

2. 4;r— 33^=1 
Zx — 4y=6 

/^Mj. : X =^ y ^= ( Score 3, 2, 1, or 0) 

3. 5;r+9y=28 
7;»r+33;=:29 

^nj. : x= y= (Score 3, 2, 1, or 0) 

Matching Equations and Problems Test: 

This test was designed to measure the ability to translate verbal 
statements of problems into algebraic symbolism. In form it is 
a matching test and has thus the advantage of isolating the task 
of translation from other factors. It has the added merit of pre- 
senting a familiar process in a novel form. The facility with 
which the subject can cope with the new situation affords some 
indication of the degree of mathematical intelligence he possesses 
rather than a measure of efficiency in a habitual method of work- 
ing. The score equals the number of problems correctly matched. 

Page 1 

Name Date 

An Equation is a short method of writing a problem. Here is a Prob- 
lem and an Equation which stands for it. 

Problem: If 7 is subtracted from a certain number, the remainder is 
13; what is the number? 

Equation: x — 7=13 

On the other side of this sheet there are 10 Problems and 10 Equations 
which stand for them. Pick from the 10 Problems the one which is 
represented by the first Equation and write its number in COLUMN 1 
opposite the Equation. Do the same for the other Problems and Equa- 
tions. 



General Conditions of the Present Investigation 21 

Page 2 

Instructions: In column 1 write opposite each Equation the number of 
the problem which it stands for. Do not write any num- 
ber twice. Write only one number opposite each letter. 
Omit no number. Do not find the answers to the prob- 
lems. Only pair the Equations and Problems. 

Problems : 

1. I had $20 in my purse when I went down town and $6 when I re- 

turned. How much did I spend? 

2. I earned $6 to-day and now have $20. How much did I have this 

morning? 

3. If six times a certain number is 20, what is the number? 

4. Each member of a class of 20 buys a copy of a book. If the class 

spends $6, how much is the book per copy? 

5. Find a number such that one-sixth of it equals 20. 

6. John and Mark both have marbles, but John has six more than 

twice as many as Mark. If John has 20 marbles, how many has 
Mark? 

7. Six less than twice a certain number is 20, what is the number? 

8. Find a number such that if the number is subtracted from 20, the 

result obtained is the same as if 6 had been added to the number. 

9. I have twice as much money as John has. If I spend $20 and he 

spends $6 we will have the same amount. How much money has 
John? 
10. Find a number such that if 20 is subtracted from twice the number 
the result is 6. 

Column 1 





Equations 


A. 


20—x=6+x 


B. 


Ar+6— 20 


C. 


2;r-|-6=20 


D. 


6jr-20 


E. 


2;r— 20=:;r-^ 




X 


F. 


— = 20 




6 


G. 


2;r-6=20 


H. 


20-;r=6 


I. 


20;r=6 


J. 


2;r— 20=6 



Matching Nth Terms and Series Test: 

The material for this test consists in a group of arithmetical 



22 Tests of Mathematical Ability and Their Prognostic Value 



progressions and of corresponding formulae. These are arranged 
in haphazard order and the subject has to match them correctly. 
The nature of the test was carefully explained, much time being 
spent in making certain that it was understood. Each formula 
correctly matched was awarded 1 mark. A series of 12 for- 
mulae and corresponding arithmetical progressions was given. 
This was followed immediately by a second series of 20 formulae, 
and 20 progressions in the case of the Horace Mann group. 



Page 1 



Name. 



Date. 



DIRECTIONS 

If in the Formula 2n we let M=first 1, then 2, then 3, then 4, then 5, 
then 6, then 7, we shall get first 2, then 4, then 6, then 8, then 10, then 

12, then 14; that is the Series of numbers 2, 4, 6, 8, 10, 12, 14 

Similarly if in the Formula 5« — 1 we again let n=l, 2, 3, 4, 5, 6, 7 in turn, 
we shall get the Series 4, 9, 14, 19, 24, 29, 34 



Therefore 



Series 



Formula 



2, 4, 6, 8, 10, 12, 14 is obtained from 2« 

4, 9, 14, 19, 24, 29, 34 is obtained from 5w— 1 

On the other side of this sheet there are twelve such Series and twelve 
Formulae from which they were obtained by letting »=first 1, then 2, 
then 3, then 4, then 5, then 6, then 7. 

The Series and Formulae have to be paired. Pick from the 12 Series 
the one which is obtained from the first Formula and write in the empty 
column, called Column 3 its number opposite the Formula. Do the same 
for the other Series and Formulae. 

Page 2 
Remember: Write in Column 3 opposite each Formula the number of 
the Series that is obtained from it. 
Write only one number opposite each Formula. 
Do not write any number twice. 



(1) 


3 


(2) 


1 


(3) 


6 


(4) 


6 


(5) 


6 


(6) 


1 


<7) 


3 



Series 

7 11 15 19 23 27 

7 13 19 25 31 37 

11 16 21 26 31 36 

12 18 24 30 36 42 
7 8 9 10 11 12 
3 5 7 9 11 13 
6 9 12 15 18 21 



Formulae Column 3 

n+5 

3n 
4n— 1 
5w+l 

6n 
n— 1 
n+2 



General Conditions of the Present Investigation 23 



(8) 1 2 3 4 5 6 

(9) 6 10 14 18 22 26 30 
ilO) 3 6 9 12 15 18 

(11) 3 4 5 6 7 8 9 

(12) 2 9 16 23 30 37 44 

(1) 8 16 24 32 40 48 .... 

(2) 8 14 20 26 32 38 .... 

(3) 8 17 26 35 44 53 .... 

(4) 4 12 20 28 36 44 .... 

(5) 4 7 10 13 16 19 .... 

(6) 8 9 10 11 12 13 .... 

(7) 8 19 30 41 52 63 .... 

(8) 8 23 38 53 68 83 .... 

(9) 8 11 14 17 20 23 .... 

(10) 4 6 8 10 12 14 .... 

(11) 4 11 18 25 32 39 .... 

(12) 8 10 12 14 16 18 .... 

(13) 8 18 28 38 48 58 .... 

(14) 8 15 22 29 36 43 .... 

(15) 8 20 32 44 56 68 .... 

(16) 8 13 18 23 28 33 .... 

(17) 8 22 36 50 64 78 .... 

(18) 8 12 16 20 24 28 .... 

(19) 4 9 14 19 24 29 .... 

(20) 8 21 34 47 60 73 .... 

Interpolation Test: 



2m—1 
3n — 3 
7n— 5 
6n— 5 
4w+2 

Formulae 
n+7 
lOn— 2 
2wH-2 
2n4-6 
12w— 4 
5w— 1 
7m— 3 

8n 
3m+1 
Sn-A 
9»— 1 
14m^-6 
3n+5 
15n-7 
4w+4 
11m— 3 
7m+1 
13n— 5 
6n-{-2 
5m+3 



Column 3 



The material for this test consists in arithmetical series from 
which certain steps have been omitted and which have to be 
replaced. 

The test was introduced in the hope that it would give some 
indication of the pupil's ability to analyze numerical or symbolic 
data, to perceive a general rule implicit in them and to apply the 
principle so derived. One mark was given for each blank cor- 
rectly filled. The blanks increase in number as the series grows 
in difficulty. The test was extended in length for the Horace 
Mann group. 

Page I 
Name Age 

Directions: Do not turn this page until the signal is given! 1 
Stop working at once when you hear stopl ! 



24 Tests of Mathematical Ability and Their Prognostic Value 

The following is a series of numbers, in which each num- 
ber follows the one before it according to a rule. 
2 4 6 8 10 12 14 16 
Thus each number is got from the one before it by adding 2 to it. 
If any of the numbers in the series is missing, it is possible to replace it. 

For example, in the series: 2 4 — 8 10 — 14 

the missing numbers are : 6 and 12 

Similarly, in the series: 5 10 — 20 25 — 35 

the missing numbers are: 15 and 30 

Similarly, in the series: 14 — 10 — — — 22 

the missing numbers are : 7, 13, 16, and 19 

On the other side of this page there are series similar to those above, 
from which certain numbers are missing. You must supply the missing 
numbers. Your score depends on the number of blanks correctly filled. 

Page 2 
(1) 



(A) 


1 


3 


5 


7 


— 


11 


13 


15 


17 





21 




(B) 


1 


5 


9 


13 


— 


21 


25 


29 


33 


— 


41 




(C) 





3 


6 


9 


— 


15 


18 


21 


24 


— 


30 




(D) 


1 


8 


15 


— 


29 


36 


43 


— 


57 


64 


71 




(E) 


7 


13 


19 


25 


— 


37 


43 


— 


55 


61 


67 




(F) 


5 


13 


— 


29 


Z7 


45 


53 


61 


— 


77 


85 




(G) 


3 


12 


21 


— 


39 


48 


57 


— 


75 


84 


93 




(H) 


2 


— 


8 


— 


14 


— 


20 


— 


26 


— 


32 




(J) 





— 


8 


— 


16 


— 


24 


— 


32 


— 


40 




(K) 


7 


— 


15 


— 


23 


— 


31 


— 


39 


— 


47 




(L) 


3 


— 


— 


18 


— 


— 


33 


— 


— 


— 


53 




(M) 


4 


— 


— 


13 


— 


— 


22 


— 


— 


— 


34 




(N) 


5 


— 


— 


— 


ZZ 


— 


— 


— 


61 


— 


— 




(P) 


8 


— 


— 


— 


28 


— 


— 


— 


48 


— 


— 




(Q) 


— 


— 


13 


— 


— 


— 


25 


— 


— 


— 


37 




(R) 


— 


— 


13 


— 


— 


— 


29 


— 


— 


— 


45 




(S) 


2 


— 


— 


— 


— 


Z7 


— 


— 


— 


— 


72 




(T) 


11 


— 


— 


— 


— 


66 


— 


— 


— 


— 


121 




(U) 


7 


— 


— , 


— 


— 


— 


43 


— 


— 


— 


— — 


- 79 


(V) 


7 










(la) 


31 










- 55 


(A) 


1 


8 


15 


22 


— 


36 


43 


50 


— 


64 


71 




(B) 


3 


7 


11 


15 


— 


23 


27 


31 


— 


39 


43 




(C) 


4 


10 


16 


22 


— 


34 


40 


46 


— 


58 


64 




(D) 


6 


10 


14 


— 


22 


26 


30 


— 


38 


42 


46 




(E) 


9 


20 


31 


— 


53 


64 


75 


— 


97 


108 


119 




(F) 


5 


17 


29 


— 


53 


65 


77 


— 


101 


113 


125 





General Conditions of the Present Investigation 25 

(G) 8 17 26 — 44 53 — 71 80 89 98 

(H) — 16 — 32 — 48 — 64 — 80 

(J) 7-13-19-25-31-37 

(K) 2 — 16 — 30 — 44 — 58 — 72 

(L) 1 _ — 7 - - 13 — — — 21 

(M) 4 — — 31 — — 58 — — -^ 94 

(N) 7 — — - 27 - — - 47 - — 

(P) 9 - — - 57 - - - 105 - - 

(Q) — — 13 — — — 29 — — — 45 

(R) __ 16 — _ — 44 — — — 72 

(S) 3 — — — — 33 — — — — 63 

(T) 2 — — — — 47 — — — — 92 

(U) 6 — — — — — 72 — — — — — 138 

(V) 5 __ — _ — 83 — — — — — 161 



Missing Steps in Series Test: 

This test is similar to the previous one. In this case, how- 
ever, the examples given involve the four common arithmetical 
processes of addition, subtraction, multiplication, and division. 
A preliminary test containing one example of each type, was first 
given. The score depended, as in the Interpolation Test, upon 
the number of blanks correctly filled. 

Page 1 
Name Date 

DIRECTIONS 

Each of the lists of numbers below is a Series, in which each number 
follows the one before it according to a rule. Thus in the 

Senes 5 10 15 20 25 30 

each number is obtained from the one before it by adding 5 to it. 
(5+5=10; 10+5=15; 15+5=20; and so on.) Similarly in 

Series 78 72 66 60 54 48 

each number is obtained from the one before it by subtracting 6 from it. 
(78—6=72; 72—6=66; 66—6=60; and so on.) Similarly in 

Series 2 6 18 54 162 486 

each number is obtained from the one before it by multiplying it by 3. 
(2X3=6; 6X3=18; 18X3=54; and so on.) Similarly in 

Series 128 64 32 16 8 4 

each number is obtained from the one before it by dividing it by 2. 
(128-^2=64; 64-^2=32; 32-^2=16; and so on.) 

The Series of numbers on the other side of this sheet are obtained in 



26 Tests of Mathematical Ability and Their Prognostic Value 

similar ways. You are to find the rule for each Series and so supply 
the missing numbers, as in the examples below. 

Examples 



Series 



(a) 


5 


10 


15 


20 


25 


30 


(b) 


78 


72 


66 


60 


54 


48 


(c) 


2 


6 


i8 


54 


162 


486 


(d) 


128 


64 


32 


16 


8 


4 
Page 2 


(1) 


1 


3 


— 


7 


9 


11 


(2) 


16 


13 


— 


7 


4 


1 


(3) 


2 


4 


— 


16 


32 


64 


(4) 


1 


8 


— 


22 


29 


Z^ 


(5) 


32 


16 


— 


4 


2 


1 


(6) 


1 


4 


— 


64 


256 


1024 


(7) 


6250 


1250 


— 


50 


10 


2 


(8) 


44 


36 


— 


20 


12 


4 


(9) 


7 


13 


— 


25 


31 


Z1 


(10) 


7 


14 


— 


56 


112 


224 


(11) 


26 


21 


— 


11 


6 


1 


(12) 


1701 


567 


— 


63 


21 


7 



Inference with Symbols Test: 

The object of this test was to ascertain how far the ability to 
manipulate symbols with ease and precision correlates with math- 
ematical ability. For this purpose five common symbols were 
chosen, of which only one, the sign for "equals," was familiar to 
the subjects. The task was to make inferences with regard to 
the relation of a pair of terms, when information about their 
relations with mediating terms had been given. Several illus- 
trative examples were shown before the test was applied. Two 
series were administered, forming a scale of increasing difficulty. 
The score equalled the number of correct inferences made. 

This test was considerably extended in the case of the Horace 
Mann group both as regards time and difficulty. 



Page 1 



Name 



Date. 



Number. 



DIRECTIONS 

Using the facts under GIVEN FACTS, fill in the blank spaces under 
FILL IN with >, <, =, > or «<, whichever gives the true con- 
clusion. In any case where none of the symbols can be correctly used, 
make a in the blank space. 



General Conditions of the Present Investigation 27 



> means "greater than" 
<C means "less than" 

= means "equal to" 

> means "not greater than" 
< means "not less than" 

Read this illustration, which will show you what is to be done. 
GIVEN FACTS FILL IN 



therefore a 
therefore a 



> 
> 



(1) 



Page 2 



REMEMBER 


> means 


"greater than" 






< means 


"less 


than" 






= means 


"equal to" 






3> means 


"not 


greater than" 






< means 


"not 


less than" 






— means 


"where none of the other 








i 


symbols fit" 


GIVEN FACTS 


FILL IN 




1. a = b = c 


therefore 


a 


c 




2. a = b > c 


therefore 


a 


c 




3. a <b =z c 


therefore 


a 


c 




4. a=:b < c 


therefore 


a 


c 




5. a > b = c 


therefore 


a 


c 




6. a <b < c 


therefore 


a 


c 




7. a> b > c 


therefore 


a 


c 




8. a <b > c 


therefore 


a 


c 




9. a> b > c 


therefore 


a 


c 




10. a <b > c 


therefore 


a 


c 




11. a <b < c 


therefore 


a 


c 




12. a > b < c 


therefore 


a 

(la) 


c 




(1) a> b = 


c = d therefore a 




d 


(2) a = b = 


c ^ d therefore a 




d 


(3) a=^b < 


c =^ d therefore a 




d 


(4) a>b> 


c = d therefore a 




d 


(5) a = b < 


c <C d therefore a 




d 


(6) a> b = 


c = d therefore a 




d 


(7) = fo = 


c > rf therefore a 




d 


(8) a = b < 


c = d therefore a 




d 


(9) a > & > 


c = d therefore a 




d 



28 Tests of Mathematical Ability and Their Prognostic Value 



(10) 


a> b> c> d 


therefore 


a 


d 


(11) 


c = & < c < d 


therefore 


a 


d 


(12) 


a -^ b <^ c <t d 


therefore 


a 


d 


(13) 


a <b < c > d 


therefore 


a 


d 


(14) 


a <b > c <d 


therefore 


a 


d 


(15) 


a> b < c < d 


therefore 


a 


d 


(16) 


a<b> c> d 


therefore 


a 


d 


(17) 


a> b> c < d 


therefore 


a 


d 


(18) 


o> b < c > d 


therefore 


a 


d 


(19) 


a = d > c < c? 


therefore 


a 


d 


(20) 


a> b = c < d 


therefore 


a 


d 


(21) 


a <b = c > d 


therefore 


a 


d 


(22) 


a>b> c> d 


therefore 


a 


d 


(23) 


a <b < c <d 


therefore 


a 


d 


(24) 


a < fe < c < rf 


therefore 


a 


d 


(25) 


a> b < c > d 


therefore 


a 


d 


(26) 


a<b =: c > d 


therefore 


a 


d 


(27) 


a <b < c < d 


therefore 


a 


d 


(28) 


a>'b > c> d 


therefore 


a 


d 


(29) 


a <b > c> d 


therefore 


a 


d 


(30) 


< i? < c > c? 


therefore 


a 


d 


Geometry Test: 









The material for this test consists in a series of geometrical 
principles together with a number of geometrical problems, whose 
solution depends upon the former. The task is to solve the prob- 
lems with the aid of the principles. In order to acquaint the 
pupils with the requirements of the test a preliminary problem 
was first given and its solution demonstrated: this brought to 
light any cases of misunderstanding of the directions. The 
method of grading the test is indicated after each problem. 

Page 1 
.THIS SHEET IS FOR REFERENCE ONLY 
Facts Given as True 



^ 




(1) All right angles have 90 degrees. 

This is a right angle. 

(2) All straight angles have 180 degrees. 

This is a straight angle. 

(3) All the angles of a triangle added together 

equal 180 degrees. 
Thus in triangle AKL, angle A, angle 
K and angle L added together equal 180 
degrees. 



General Conditions of the Present Investigation 29 







(4) Tzc'o triangles are equal if two sides and 
the angle between them are equal. 

Thus the triangle AHL equals triangle 
DEF since side AH equals side DE and 
side HL equals side EF, and angle H 
equals angle E. 

(5) Two triangles are equal if two angles and 
the side between them are equal. 

Thus triangle KLM equals triangle 
TVN, since angle K equals angle T and 
angle L equals angle V and side LK 
equals side TV. 

(6) An isosceles triangle has two sides equal 
and the angles opposite them equal. 

Thus FTH is an isosceles triangle, since 
FT equals TH and angle F equals 
angle H. 

(7) The sides of a square are equal, and all 
its angles are right angles. 
Thus AVTF is a square, since AV equals 
VT equals TF equals FA, and angles 
A, V, T and F are right angles. 

(8) // two lines are equal, their halves are equal. 
Thus AC equals DF, therefore BC (half 
of AC) equals EF (half of DF). 

(9) // two angles are equal their halves are 
equal. 

Thus angle FHN equals angle KLM, 
therefore THN (half of angle FHN) 
equals angle VLM (half of angle KLM). 
Directions: Taking the facts (1), (2), up to (9) as true, do the work 
required on the other sheet and write your answer in the space re- 
served. Do not take anything for granted not given in (1), (2), etc. 
above or under Given on the other sheet. First do 1, then do 2, and 
so on. In every case when you use any of the facts above (1), (2), 
etc. in your work, write the number to show what fact it is you use. 



\ 



vl\. 




Example 




Given: Angle Lr=60 degrees. 

The triangle is isosceles. 
Prove: Angle F= ? degrees. 



Answer : 

The triangle is isosceles (Given). 
Therefore angle L equals angle F by fact (6). 
Angle L equals 60 degrees (Given). 
Therefore angle F equals 60 degrees. 



30 Tests of Mathematical Ability and Their Prognostic Value 



Answer: 



Given: Angle X=:ZO degrees. 
Prove: Angle Z^ ? degrees. 



Page 2 



Angles X and Z=180 degrees (Score 1) by fact 

2 (Score 1). 
Therefore Angle Z=: 150 degrees (Score 1). 



.^. 



Answer 



Given: Angle L is a right angle. 
Prove: Angle i^=? degrees. 
Angle M=30 degrees. 

Angle L=9Q degrees (Score 1) by fact 1 (Score 

1). 
Angles K, L, and M=180 degrees (Score 1) by 

fact 3 (Score 1). 
Therefore /^=60 degrees (Score 1). 



3. 



Answer : 



Given: The triangle is isosceles. 

Angle P=:30 degrees. 
Prove : Angle Q=: ? degrees. 

Angles P and Q are equal (Score 1) by fact 6 

(Score 1). 
Therefore Angle 0=30 degrees (Score 1). 



Answer : 



Given : The triangle is isosceles. 

Angle /^=:30 degrees. 
Prove: Angle X= ? degrees. 

Angles A and D are equal (Score 1) by fact 6 

(Score 1). 
Angles X and D together=:180 degrees (Score 1) 

by fact 2 (Score 1). 
Therefore Angle Z=150 degrees (Score 1). 




Given: The triangle is isosceles. 

The line from ^ to Z) is drawn so as 
to make angle jB/iZ)r=angle DAC. 
Prove: The two small triangles equal 



General Conditions of the Present Investigation 31 



Answer : 



6. 




Answer ; 



Angle B=:Angle C (Score 1) by fact 6 (Score 1). 
Angle BAD= Angle DAC, Given (Score 1). 
Line BA=:Unt AC (Score 1) by fact 6 (Score 

1). 
Therefore the 2 triangles are eqtial by fact 5 

(Score 1). 



Given: The figure is a square. 

Two opposite corners are joined. 
Prove: The square is divided into equal tri- 
angles by the line drawn. 

Line ^5=Line BD (Score 1) by fact 7 (Score 

1). 
Line ^C=Line CD (Score 1) by fact 7 (Score 

1). 
Angle ^=; Angle D (Score 1) by fact 7 (Score 1). 
Therefore the 2 triangles are equal by fact 4 

(Score 1). 
(Corresponding system of marking for proof by 

fact 5.) 



Superposition Test: 



This test was developed by L. L. Thurstone, of the Carnegie 
Institute of Technology, Pittsburgh, as a measure of the ability 
to grasp spatial relations. In the present investigation it served 
two purposes. It measured the dexterity with which the sub- 
ject could apply both the principle of superposition and that of 
symmetry. It consists essentially of pairs of symmetrical paral- 
lelograms, each with one side on the same straight, long, black 
line and each adjoining a third parallelogram of corresponding 
design, and similarly with one black edge, but such that it can be 
superposed upon only one of the adjoining parallelograms. 
This third parallelogram which has a small circle in one corner 
is placed in a variety of positions relative to the pair of parallelo- 
grams. The task, in this case, was to imagine the third paral- 
lelogram moved around so that it fitted one of the corresponding 
pair of parallelograms and to indicate which it was by drawing a 
small circle in the comer of the same, exactly where the circle in 
the third parallelogram would then lie. 

The score equals the number of circles correctly placed. This 



32 Tests of Mathematical Ability and Their Prognostic Value 

test was applied twice to the Wadleigh High School group and 
four times to the Horace Mann group. 

Symmetry Test: 

The material for this was the same as the foregoing, while the 
method was changed. The subject had to imagine the third paral- 
lelogram picked up, turned over and placed face down with its 
black edge touching the long heavy, black line to the right. The 
card was then imagined to be moved until its edges fitted the 
edges of one or other of the two parallelograms. A circle had 
to be drawn in the corner where the circle in the third parallelo- 
gram would then lie. The score equals the number of circles 
correctly marked. This test was extended when applied to the 
Horace Mann group. 

The following examples are typical : 




^ /VV\ 

r7v\ 



Matching Solids and Surfaces: 

The purpose of this test was similar to that of the two preced- 
ing with this difference that it was expressly directed towards 
estimating the facility with which tri-dimensional relations were 
intuitively grasped. In form it resembles a matching test, the 
subject having to name all the solids from which the given sur- 
faces could be obtained by a single cross-section and to indicate 
the nature of the transection necessary. For each solid cor- 
rectly named a score of 1 was given ; for each section of correct 
shape indicated an additional mark was awarded, and where a 



General Conditions of the Present Investigation 33 



section of correct size as well as shape was indicated two addi- 
tional marks were assigned. 

REFERENCE SHEET: 
These are drawings of Solids. 

(1) (2) 





Fig. 1 is 2 inches high and its base is 1 square inch in size. 
Fig. 2 is 1 cubic inch in size. 



These are drawings of Surfaces. 
(3) 



(4) 




Fig. 3 is half an inch square. It is obtained by cutting Solid 1 straight 
through once through the points A, B, C and D. 

Fig. 4 is one inch high and one and two-fifths inches broad and is 
obtained by cutting Solid 2 straight through once through the points 
A, B, C and D. 

On the other sheet there are seven drawings of Solids and below 
these there are seven drawings of Surfaces obtained by cutting these Solids 
straight through once- with a flat knife. 
Solid I. Altitude 2 inches ; Base 1 inch square. 
Solid II. Altitude 2 inches; Base, diameter 1 inch. 
Solid III. Altitude 2 inches; Base 1 inch on each side. 
Solid IV. Diameter 1 inch. 

Solid V. Altitude 2 inches; Base 1 inch square. 
Altitude 2 inches ; Base, diameter 1 inch. 
Major axis one and one-half inches; Minor axis 1 inch. 



Solid VI. 
Solid VII. 



34 Tests of Mathematical Ability and Their Prognostic Value 

DIRECTIONS 

Examine Surface 1. Decide on all the Solids from which it can be 
obtained by cutting them straight through once with a fiat knife in any 
direction. Write in the empty space below the Surface the numbers of 
the Solids from which it can be obtained. Also show how the Solids 
must be cut by lettering the points on the edges of the Solids through 
which the knife must pass. Write these letters after the corresponding 
number of the Solid in the space below. Do the same for Surfaces 
2, 3, 4, 5, 6 and 7. 

REMEMBER: (1) Give the points through which the knife must 
pass; do not give the points which outline the re- 
quired Surface. 

(2) The points through which the knife passes must 
give the Surface correct in both shape and size. 

(3) The same Surface may be obtained from several 
Solids. 

(4) No Surface should be obtained more than once 
from the same Solid. 

(5) One point has only one letter-name. 

(6) The effect of perspective. 

[See cut opposite page] 

Geometrical Definitions Test: 

This a modified form of a test by Winch. ^ After reading 
carefully a series of illustrative examples, the subjects had to 
write definitions for geometrical figures of various kinds, v^hich 
were shown. The ability to analyze common and also differen- 
tiating features and to generalize from spatial data was meas- 
ured by this test. The method of marking followed was that 
used by its originator. 

3 See Winch, W. H., Inductive versus Deductive Methods of Teaching : 
An Experimental Research, Baltimore, 1913. 



I General Conditions of the Present Investigation 35 




^a 



O 










^ 



36 Tests of Mathematical Ability and Their Prognostic Value 

GEOMETRICAL DEFINITIONS TEST 
Preliminary Examples 




1. These are Squares, therefore a Square is a figure with 4 equal 
straight lines as sides and 4 equal angles which are right angles. 




2. These are Rectangles, therefore a Rectangle is a figure bounded 
by 4 straight lines, of which the opposite sides are equal and parallel 
and whose angles are all equal and right angles. 





3. These dotted lines are Secants, therefore a Secant is a straight line 
that cuts the circumference of a circle in 2 points. 



General Conditions of the Present Investigation 37 

Geometrical Definitions: 





1. These are Rhomboids; give a complete definition of a Rhomboid. 






2. These are regular Pentagons; give a complete definition of a regu- 
lar Pentagon. 





<3 



3. These are Trapezoids; give a complete definition of a Trapezoid. 






4. These dotted lines are Tangents; give a complete definition of a 
Tangent to a circle. 



38 Tests of Mathematical Ability and Their Prognostic Value 
Reasoning Test: ^' 

This test was devised on lines suggested by Thorndike. It 
consists of a ladder-like arrangement of inferences, of gradually 
growing complexity and difficulty. It was designed to measure 
the ability to seize the relevant elements in a complex and abstract 
situation and to respond only to them. The score depended upon 
the number of correct inferences drawn. 



Page 1 



Name. 



Date. 



DIRECTIONS 



On the other side of this sheet fill in the blank spaces under fill in with 
conclusions which can be correctly drawn from the facts stated under 
given facts. 

Read this illustration which will show what is to be done. 

Given Facts Fill In 



R is shorter than S 
T is longer than 5 
R is longer than V 



therefore 
therefore 



T is longer than R. 
V is shorter than S. 













Page 2 




Given Facts 








Fill In 


(1) 


P is longer than Q 












R is shorter than Q 


therefore 


P 


is.. . 


R 


(2) 


M is younger than N 












K is older than N 


therefore 


K 


is.. . 


L 




M is older than L 


therefore 


N 


is... 


L 


(3) 


M is richer than 












is as rich as P 


therefore 


M 


is... 


K 




K is poorer than P 


therefore 


N 


is. . . 


M 




N is poorer than K 










(4) 


Z is thicker than X 


therefore 


X 


is.. . 


H 




H is as thick as Z 


therefore 


Y 


is.. . 


H 




V is thicker than H 


therefore 


X 


is. . . 


V 




V is thinner than Y 


therefore 


Z 


is. . . 


Y 


(5) 


D is greater than B 


therefore 


B 


is... 


A 




B is equal to E 


therefore 


D 


is.. . 


F 




E is greater than F 


therefore 


E 


is. . . 


A 




C is less than F 


therefore 


B 


is... 


C 




A is greater than D 


therefore 


A 


is... 


F 



General Conditions of the Present Investigation 39 

(6) ^ is higher than G therefore ^ is D 

B is equal to E 

D is lower than G therefore F is B 

C is higher than G 

G is lower than B therefore H is E 

B is hegher than H 

A is lower than H therefore D is C 

F is equal to H 

E is higher than G therefore E is F 

Graded Problems in Arithmetic: 

This test was constructed by Thorndike. It is self-explanatory. 
One mark was given for each answer, if absolutely correct. 

Name Date 

GRADED PROBLEMS (1) 

Find how long Mary was allowed to play on each of these days. 

Answers 

1. Monday. It is 4.10 P. M. Supper is at 6 o'clock. 
Mother says "you may play half the time from now 
till supper time." 



2. Wednesday. It is 4.05 P. M. Mother says "if 
you help me for half an hour now, and for 10 minutes 
before supper you may play the rest of the afternoon." 

3. Friday. Mother says you may play 2 minutes 
for every three problems you solve, and five minutes 
more for every problem you solve correctly. Mary 
solved 15 and has all but one right. 



The rest of these problems all ask the same question: how many 
minutes is it from the time John begins to pump until the tank is filled? 
The tank holds 120 gallons and is supposed always to be empty when 
John begins to work. 

Answers 

4. John pumps 2 minutes before any water reaches 
the tank. Then he pumps water into it at the rate of 
3 gallons a minute until the tank is full. 



5. John pumps 8 minutes before any water reaches 
the tank. Then he pumps water into it at the rate of 
24 gallons in 10 minutes until the tank is full. 



40 Tests of Mathematical Ability and Their Prognostic Value 

6. John pumps 1^ minutes before any water reaches 
the tank. Then he pumps for 10 minutes at the rate 
of 2.7 gallons a minute. Then the pump breaks and 
he spends 8 minutes mending it. Then he pumps at 
the rate of 3.1 gallons per minute until the tank is full. 



Thorndike Reading Tests: * 

The Thorndike Reading Scale Alpha 2 was used for the first 
measure of the ability to understand sentences and a number of 
passages of corresponding complexity for the second. The stan- 
dard method of marking was adopted * in the case of Scale Alpha 
2. For the passages, the scale of marking was 4, 3, 2, 1, 0. 

Tests of Verbal Ability 

1. Mixed Relations Test: ^ 

This is a well-known test, in which the task is to discover a 
fourth term, which stands in the same relation to the given third 
term as the second does to the first. Twenty such examples were 
presented to the Wadleigh High School girls and double that num- 
ber to the Horace Mann group. Three marks were given for a 
perfect score and two or one for partially correct solutions ac- 
cording to the degree of the correctness.^ 

2. Logical Opposites Test: '' 

In this test the subject was given a list of thirty words in the 
case of the Wadleigh High School students and one hundred 
words in the case of the Horace Mann group. The logical op- 
posite had in each instance to be written. The score for a perfect 
answer was 3, while 2 or 1 was given for efforts of a less ap- 
propriate kind according to the degree of correctness. 

* See Teachers College Record, September, 1914, November, 1915, and 
January, 1916. 

5 See Whipple, G. M., Manual of Mental and Physical Tests, 2d ed., 
1914, Pt. 2: 89-94, for a description of the use that has been made of 
the Mixed Relations Test. 

[See page 41 for footnote 6] 

■^ For the previous application of this test the reader is referred to 
the same source, 79-89. 



General Conditions of the Present Investigation 41 

3. Trabue Language Scales: ^ 

These consist of a series of sentences from which certain words 
have been omitted. The sentences are graded in difficulty and 
standardized. Scales L and M were given to the Wadleigh High 
School group and these together with Scales J and K to the 
Horace Mann pupils. The method of scoring used was that fol- 
lowed in standardizing the scales. 



terial used in the Mixed Relations Test was 


the following 


eye — see 


ear — 




Monday — Tuesday 


April- 




do— did 


see — 




bird — sings 


dog — 




hour — minute 


minute — 




straw — hat 


leather — 




cloud — rain 


sun — 




hammer — tool 


dictionary — 




uncle — aunt 


brother — 




dog — puppy 


cat- 




little— less 


much — 




wash — face 


sweep — 




house — room 


book- 




sky — blue 


grass — 




swim — water 


fiy- 




once — one 


twice — 




cat — fur 


bird- 




pan — tin 
buy — sell 


table— 




come — 




oyster — shell 


banana — 




past — present 


present — 




come — came 


go — 




north — south 


far- 




mend — clothes 


bake— 




lily — flower 


oak- 




ton — pound 


pound — 




elbow — arm 


chin — 




pea — pod 


nut — 




bell — rings 


clock- 




deep — valley - 


high- 




growls — dog 


roars — 




brick — wall 


page — 




lathe — machine 


hammer — 




pencil — lead 
high — low 


book — 




up — 




sheep — lamb 


dog — 




kettle — brass 


cup — 




Thursday — Friday 


June — 




build — house 


paint — 




one — single 


two — 




mice — cat 


worms — 




London — England 


Paris- 




A church organ — banjo 


Hamlet— 





A corresponding duplicate series was arranged. 

8 See Trabue, M. R., Completion-Test Language Scales, Teachers Col- 
lege, Columbia University Contributions to Education, No. 77. 



CHAPTER III 
THE ANALYSIS OF MATHEMATICAL ABILITY 

From the foregoing description one may judge to what extent 
the tests are representative of the activities involved in high school 
mathematics and how far the essential data for the theoretical 
analysis of mathematical intelligence have been secured. What- 
ever their Hmitations and defects, it can hardly be doubted that 
these tests of algebraic and geometrical abilities give valuable in- 
formation about the intellectual efficiency of pupils in first-year 
mathematics and further independent objective evidence will be 
presented later to show that they actually do measure important 
elements in the mental equipment of the prospective student of 
the subject. Together with the tests of language ability, which 
give auxiliary aid in interpreting results, they furnish a workable 
instrument for experimental analysis and research. For purposes 
of practical diagnosis ease in administration is essential. The 
tests must be convenient as well as typical and comprehensive. 
Their practicability may likewise be judged by the preceding de- 
scription, together with the detailed instructions to be found in 
the Appendix. 

It is further necessary, if the tests proposed are to be considered 
satisfactory, that they should be reliable measuring rods of the 
abilities investigated. We have an objective indication of the 
reliability of a test, when two distinct series of measurements by 
the same test of the same group give similar results. Thus for 
a test to be scientific and trustworthy, the relative positions of 
the individuals examined should be the same on every applica- 
tion. On the other hand, if chance is exercising a preponderating 
influence upon the results, slight correspondence between several 
trials will be found. The amount of such correspondence be- 
tween any two applications can be given precise quantitative ex- 
pression in the coefficient of correlation derived from two inde- 

42 



The Analysis of Mathematical Ability 43 

pendent sets of measures. This reliability coefficient is condi- 
tioned in part by the number of cases examined. If a sufficiently 
representative sample has been tested and the value of the re- 
liability coefficient obtained is small, the test should obviously be 
reconstructed. Whenever the coefficient is less than .60 and pro- 



TABLE I 

Reliability Coefficient for Each Test, and for Its Two Applications 
Combined. Wadleigh High School 



T r 

^1 '2 

Algebraic Computation 77 .87 

Matching Equations and Problems 63 .77 

Matching Nth. Terms and Series .67 .80 

Interpolation 71 .83 

Missing Steps in Series 75 .86 

Inference with Symbols 23 .38 

Geometry 66 .80 

Superposition 84 .91 

Symmetry 92 - .96 

Matching Solids and Surfaces 35 .52 

Geometrical Definitions 31 .47 

Mixed Relations 42 .59 

Logical Opposites 85 .79 

Trabue Language Scales 46 .63 

Thorndike Reading Tests 50 .67 

Reasoning 43 .60 

Arithmetic Problems 46 .63 



Note: 

r^ is the Reliability Coefficient or coefficient of correlation between 
two applications of the tests. 

r^ is the Reliability Coefficient for the two applications of the tests 
combined, r equals 



It measures the extent to which the amalgamated results of the two 
applications would correlate with a similar amalgamated pair of two other 
applications of the same test. See Brown, William, The Essentials of 
Mental Measurement. Cambridge, 1911:101-102. 



44 Tests of Mathematical Ability and Their Prognostic Value 

vided the number of cases is sufficiently large, the test ought to 
be improved. In many such instances it may only require ex- 
tension. 

The reliability coefficients for each test were first calculated for 
the Wadleigh High School group. They are summarized in 
Table I. The numerical data upon which they are based, as 
also all the other results of this investigation, are recorded in 
the Appendix in Tables XXXIV and XXXV in which the original 
scores are given. 

Six of the coefficients are under .50 and even when allowance 
was made for their attenuation by the different conditions in the 
two applications and the length of time intervening, to which we 
have already referred, it seemed desirable that the tests should be 
lengthened and improved, wherever that was practicable, before 
further application was made. 

With a view to determining from the ascertained reliability 
coefficients the number of applications of any particular test, 
which would be necessary to give an amalgamated result of ap- 
proximately perfectly reliability, the formula suggested by William 
Brown ^ was used, 



nr^ 

rn — 



l + {n-l)r, 

where n represents the number of applications of a test and r^ is 
the coefficient of correlation between any two applications. 

Besides being prolonged in accordance with this guiding rule, 
several of the tests were amended as regards method of applica- 
tion and increased in difficulty, so that a scale better fitted to 
differentiate degrees of ability was evolved. Owing to lack of 
time, all of the extensions prepared could not be applied. While 
the following tests were given in identical form to both the Wad- 
leigh and the Horace Mann groups, — Algebraic Computation, 
Matching Equations and Problems, Missing Steps in Series, Geom- 
etry Test, Matching Solids and Surfaces, Geometrical Defi- 

1 Brown, William, The Essentials of Mental Measurement, Cambridge, 
1911, 101-102. 



The Analysis of Mathematical Ability 45 

nitions, Reasoning Test, Arithmetic Problems, and the Thorn- 
dike Reading Tests, the eight remaining tests were adminis- 
tered in the extended form in which they now appear. 

The reHability coefficients obtained from the second appHca- 
tion are presented in Table 11. 



TABLE II 

Reliability Coefficient for Each Test, and for Its Two Applications 
Combined. Horace Mann School 

^ ''a 

Algebraic Computation 79 .88 

Matching Equations and Problems 61 .75 

Matching Nth Terms and Series 81 .89 

Interpolation 94 .97 

Missing Steps in Series 70 .82 

Inference with Symbols 82 .90 

Geometry 75 .86 

Superposition 82 .90 

Symmetry 96 .98 

Matching Solids and Surfaces 70 .82 

Geometrical Definitions 84 .91 

Mixed Relations 79 .88 

Logical Opposites 58 .73 

Trabue Language Scales 33 .49 

Thorndike Reading Tests 57 ,73 

Reasoning 73 ,85 

Arithmetic Problems 81 .76 



Note: 

r^ is the Reliability Coefficient or coefficient of correlation between two 
applications of the tests, 

fg is the Reliability Coefficient for the two applications of the tests 

combined, r^ equals 

l+r. 

It measures the extent to which the amalgamated results of the two 
applications would correlate with a similar amalgamated pair of two 
other applications of the same test. See Brown, William, The Essentials 
of Mental Measurement, Cambridge, 1911 : 101-102. 



46 Tests of Mathematical Ability and Their Prognostic Value 

It will be seen that in all the tests of mathematical abilities 
whose reliability coefficients on the former application were con- 
spicuously low, there is a marked increase in reliability, while of 
the three tests of verbal ability, which were doubled or more than 
doubled in length, only one, Mixed Relations, shows any improve- 
ment. The causes for this can be traced to the unfavorable ex- 
perimental conditions. Whereas all the tests in algebra and geom- 
etry were made in a regular class period of forty minutes dura- 
tion, these three tests of language ability were applied in a short 
twenty-minute period from 9 to 9:20 a. m., in which the pupils 
are usually given an opportunity to consult with their section- 
teacher, should that be necessary. Consequently there was neither 
the same readiness nor concentration of attention that character- 
ized their behavior in the remainder of the tests. 

We know, however, that when the conditions of experimenta- 
tion are favorable, these tests furnish adequate reliability coef- 
ficients and as they are primarily introduced in this study not for 
use in isolation, but en masse as a measure of language efficiency, 
the reliability coefficients yielded by their amalgamated results 
are sufficiently high to be satisfactory. 

The increase in the amount of the reliability coefficients of the 
mathematics tests in the second as compared with the first group 
of pupils examined is largely due to the fact that irrevelant factors 
were excluded to a much greater extent. The second application 
of each test of the same mental function, for example, was usually 
given at the same hour of the following day and the time devoted 
to testing was invariably thirty-five to forty minutes. 

Having determined the amount of confidence to which the 
tests are entitled, we can now consider the extent of connection 
which they reveal between the various functions examined. This 
is best shown by the quantitative expression of correspondence in 
coefficients of correlation. The standard "Product-Moments 
method," discovered by Bravais in 1846 and demonstrated by 
Pearson in 1896 to be the most satisfactory, has been used 
throughout this study. The formula in its most convenient form 
is as follows : 

r = 



y^x^%'^' 



The Analysis of Mathematical Ability 47 

In this r is the required correlation, x and y are the deviations 
of any pair of characteristics from their respective central tend- 
encies, 2 xy is the sum of such products for all individuals, 2 x^ 
is the sum of the squares of all the values oi x, ^ y^ is the sum 
of the squares of all the values of y. Where any existing posi- 
tive relationship is observed between two traits, the coefficient as- 
sumes some value between and +1 ; where an existing inverse 
or negative relationship is found, the value of the coefficient lies 
somewhere between and — 1. The coefficient thus may have 
any value from -|-1 through to — 1, according as the relationship 
is present in some amount or absent and according as the cor- 
respondence is positive or negative in nature. 

Before we can attribute evidential value to the coefficients of 
correlation obtained by means of the above formula, however, 
it is essential to determine their probable errors due to the fact 
that only a limited sample of the total number of high school 
pupils has been examined. Some measure of the variability in 
results that we must expect, should other groups of individuals 
be tested, has to be provided. A sufficiently accurate formula for 
determining the probable error of a coefficient of correlation, 
when the number of cases is fairly large and the distribution of 
frequencies is normal has been suggested by Pearson. 

P.E. = .67449 

V n 

This defines the limits within which a coefficient may vary in 
value by accident. Its meaning may be seen from the state- 
ment that the chances are even that the true value of the coef- 
ficient r lies between the limits 

.67449(1—^2) 
r ±: 



V w 

Each coefficient of correlation will then have to be compared 
with its probable error in order to ascertain whether it demon- 
strates any actual interdependence of the functions in question. 
For a coefficient to be considered satisfactory evidence of an 



48 Tests of Mathematical Ability and Their Prognostic Value 

existing correspondence it has to be several times larger than its 
Probable Error. No coefficient less than twice as large can es- 
tablish any conclusion about the actual existence of functional 
interdependence between two abilities. In Tables III and IV, 
the Probable Errors for the Wadleigh High School group and 
the Horace Mann School group have been calculated for the 
various values of r by the formula,^ 



P.E. =z .6744898 



(1-r^) 



V w 



TABLE III 

Probable Error of the Coeffi- 
cients OF Correlation: 

Wadleigh High School (n=53) 
r P.E. 

.9 02 

.8 03 

.7 OS 

.6 06 

.5 07 

.4 08 

.3 08 

.2 09 

.1 .09 



TABLE IV 

Probable Error of the Coeffi- 
cients OF Correlation: 

Horace Mann School (n=6l) 
r P.E. 

.9 .02 

.8 03 

.7 04 

.6 06 

.5 06 

.4 07 

.3 08 

.2 08 

.1 09 



We are now in a position to examine critically the results ob- 
tained from the application of the statistical methods described 
above to the two sets of data which form the basis of this study. 
Coefficients of correlation were calculated separately for the two 
groups, since spurious correlation would have arisen, had their 
records been mingled.^ In Tables V and VI, the coefficients of 
correlation between each application of each test and a corres- 
ponding application of every other test are summarized for the 
Wadleigh High School and the Horace Mann School groups re- 

2 Winifred Gibson's "Tables for Facilitating the Computation of Prob- 
able Errors" in Biometrika, IV : 385, were used and David Heron's "Abac 
to determine the Probable Errors of Correlation Coefficients" in Bio- 
metrika, VII: 411. 

3 See Yule, G. Udny, An Introduction to the Theory of Statistics, Lon- 
don, 1916, 218-219. 



The Analysis of Mathematical Ability 49 

spectively. The coefficients found between the two applications 
of every test and age are likewise included. In Tables VII and 
VIII corresponding pairs of coefficients in the two preceding 
tables are amalgamated so that each coefficient in Tables VII and 
VIII is the average of two corresponding applications of a pair 
of tests. Table IX combines the coefficients from the two groups, 
giving double weight to the Horace Mann results in virtue of their 
greater reliability. The tables of amalgamated coefficients pre- 
sent the results in a more comprehensible form and facilitate their 
interpretation. 

These correlation tables furnish the data for an analysis of 
mathematical ability. The subtle interrelations of the complex 
capacities, which we vaguely indicate by the term mathematical 
intelligence, are already partially revealed in these figures and 
closer scrutiny will disclose more fully the nature of the corres- 
pondences that hold between the various functions involved. 



50 Tests of Mathematical Ability and Their Prognostic Value 

TABLE V 
Crude Coefficients— Wadleigh High School 







^ 


^ 


•c 


•?< 


















O 


o 






















C^ 


PM 




WD 




















TS 


(3 


C 
















•J! 

3 


s 


c 


rt 


(4 






(0 


10 


Ji2 


"5 


o 


CO 


CO 

§ 


u 


CO 






u 

w 


•5 




1 


i 






H 


H 






c 


C 


w 


Ui 


6 


g 


3 


V 














X 


JS 


^ 


u 






!^ 


Z 


1 

rt 


§ 




a 


'^ 


% 


u 


o 


bo 


bo 


bo 


be 


cS 


w 


C/J 


«> 


V 


r 


c4 


c 


a 


c 


c 


•3 


*3 


bo 


bo 


g 




U3 


^ 




•s 


^ 
u 




& 


& 


.S 


.S 


S 


J> 


bi) 


te 


rt 


rt 


c« 


(4 




+5 


M 


^ 


V 


< 


< 


S 


:^ 


§ 


S 






s 


§ 




a 

M 



1 5 

& s 

o o 



Algebraic Computation 

Algebraic Computation 

Matching Equations and Problems. 
Matching Equations and Problems. 
Matching Nth Terms and Series . . 
Matching Nth Terms and Series.. 

Interpolation 

Interpolation 

Missing Steps in Series 

Missing Steps in Series 

Inference with Symbols 

Inference with Symbols 

Geometry 

Geometry 

Superposition 

Superposition 

Symmetry 

Symmetry 

Matching Solids and Surfaces 

Matching Solids and Surfaces 

Geometrical Definitions 

Geometrical Definitions 

Reasoning 

Reasoning 

Arithmetic Problems 

Arithmetic Problems 

Mixed Relations 

Mixed Relations 

Logical Opposites 

Logical Opposites 

Trabue Language Scales 

Trabue Language Scales 

Thorndike Reading Tests 

Thorndike Reading Tests 

Age 

Age 



.54 



.38 



.16 



.07 



.37 



.34 



.61 



.49 



.54 






.11 


.02 


.34 


.38 




.23 




.26 


.24 


.07 


.23 






.17 


.01 


.16 


.11 






.04 


.23 


.37 


.26 




.04 




.37 


.34 


.02 


.17 






.30 


.49 


.24 




.23 


.30 




.61 


.34 


.01 




.37 




.04 


.01 




.01 


.15 


—.23 


.11 


.01 


.25 




.31 


.19 


.31 


.22 




.19 


.07 


.00 


.26 


.21 


.03 




.39 


.15 


.28 


.31 




.29 


.19 


.21 


.26 


.11 


.10 




.33 


.19 


.10 


.19 




.18 


.10 


.02 


.05 


.15 


—.01 




.08 


.07 


.36 


.18 




.19 


.10 


.21 


.16 


.02 


.08 




.22 


.23 


.31 


.06 




.01 


.27 


.16 


.11 


.17 


.17 




—.03 


.10 


.21 


.25 


— 


.08 


.38 


.24 


.31 


.19 


.06 




.32 


.17 


.02 


.15 




.12 


.40 


.20 


.06 


.26 


—.05 




.25 


.26 


.01 


.32 




.15 


.06 


.04 


.19 


.32 


—.11 




.26 


.16 


.28 


.29 




.26 


.15 


.31 


.28 


.30 


—.02 




.32 


.28 


.29 


.20 




.21 


.02 


.08 


.36 


.43 


—.05 




.21 


.18 


—.09 


.26 




.01 


.19 


—.17 


.13 


.27 


—.11 




.17 


—.05 


—.25 


—.17 





.00 


—.39 


—.15 


,24 


—.12 


—.15 




—.35 


—.22 



.04 
.01 



-.11 
.01 



.01 
.15 



.25 
.31 



.23 



.19 



—.02 
.14 



.14 
.10 



.24 
.17 



.19 



.26 
.24 



.03 
.39 



.02 — , 



.06 
.05 

.25 
.20 

.07 
.38 

.33 
.41 

.13 
.06 

.10 
.24 

.13 
.34 

.23 
.18 

.34 
.24 , 

—.06 I! 
39 

—.04 
23 



The Analysis of Mathematical Ability 



SI 



Table V — Continued 









V 3 


CO 

.5 


o 

!5 






in 

E 


to 

s 










'a 

o 
tn 


"cfl 


b« 


bo 




g § 

*•> 'm 

1 1 




>> 

a 


rt c« 

l/i CO 

t« bo 

1 1 


G 

1 

i 


'5 

1 

a 


bo 

*c 
8 




3 
o 

1 


1 

Ph 
o 

6 


(0 

C 

o 


V 


1 

a 
O 

H 


CO 
V 

1 

a 
O 

.1 


bo 

c 

3 


ri 


•-3 
s 


'i 

c 




a a 


g 


g 


4-> •<.' 


o 


o 


cd 


rt 


♦J 




X 


X 


'5 


bi 


*§ 


(13 


o 


o 


4) « 


^ ^ 


c^ 


c^ 






6 


(A 


^ 


'< 


< 


ii 


ii 


o 


,3 


H 


s5 


^ 


J3 


bo bo 

< < 


^ <N 


- 


CVI 


T-l (M 


"ZT 


CO 


- 


<M 


- 


<N1 


"~~ 


(M 


- 


oa 


- 


tM 


- 


eg 


y-i PQ 


.26 




.05 


.16 




.11 




.31 




.06 




.19 




.28 




.36 




.13 


—.24 


.28 


.10 




.36 


.31 




.21 




.02 




.01 




.28 




.29 




—.09 




—.25 


.11 




.15 


.02 




.17 




.19 




.26 




.32 




.30 




.43 




.27 


—.12 


.31 


.19 




.18 


.06 




.25 




.15 




.32 




.29 




.20 




.26 




—.17 


5 .10 




.01 


.08 




.17 




.06 


— 


.05 


— 


.11 





.02 




.05 




-.11 


—.15 


.29 


.18 




.19 


.01 




—.08 




—.12 




—.15 




.26 




.21 




.01 




—.00 


! .33 




.08 


.22 




-.03 




.32 




.25 




.26 




.32 




.21 




.17 


—.35 


.19 


.10 




.10 


.27 




.38 




.40 




.06 




.15 




.02 




.19 




—.39 


.19 




.07 


.23 




.10 




.17 




.26 




.16 




.28 




.18 





-.05 


—.22 


.21 


.02 




.21 


.16 




.24 




.20 




.04 




.31 




.08 




—.17 




—.15 


.12 




.21 


.01 


— 


.02 




.14 




.10 




.20 




.31 




.16 




.19 


.02 


.15 


.14 




.10 


.32 




.21 




.14 




.06 




.24 




.17 




.26 




—.24 


.05 




.20 


.38 




.41 




.06 




.24 




.34 




.18 




.24 




.39 


—.23 


i.06 


.25 




.07 


.33 




.13 




.10 




.13 




.23 




.34 




—.06 




-.04 


ii 




.52 


.37 




.14 




.32 




.09 




.23 




.01 




.07 




-.05 


—.05 


' 


.69 




.36 


.25 




.24 




.29 




.11 




.16 




—.01 




.14 




—.01 


] .69 






.42 




.38 




.06 




.23 




.26 




.17 




.25 




.23 


.06 


1.52 






.06 


.37 




.23 




.32 




.01 




—.01 




.07 




.21 




.06 


.36 




.06 






.00 




.17 




.05 




.31 




.00 




-.05 




.05 


.05 


.37 


.42 






.37 




.31 




.23 




.04 




.23 




.19 




.19 




—.26 


.25 




.37 


.37 








.06 




.43 




.30 




.31 




.36 




.32 


—.15 


.14 


.38 




.00 






—.04 




—.16 




—.01 




.22 




.14 




—.14 




.06 


.24 




.23 


.31 


— 


-.04 








.28 




.12 




.17 




.13 




.02 


—.21 


.32 


.06 




.17 


.06 








.16 




.15 




.22 




.22 




—.06 




—.05 


.29 




.32 


.23 


— 


-.16 




.16 









.01 




.13 




.16 




.12 


— .13 


1.09 


.23 




—.05 


.43 




.28 








—.09 




—.01 




.22 




—.08 




—.25 


L •" 




.01 


.04 


— 


-.01 




.15 




.09 








.33 




.01 




-.03 


.11 


1.23 


.26 




.31 


.30 




.12 




—.01 








.45 




.21 




.30 




—.13 


{ .16 




.01 


.23 




.22 




.22 




.01 




.45 








.47 




.51 


—.17 


loi 


.l7 




.00 


.31 




.17 




.13 




.33 








.23 




.22 




—.02 


; —.01 




.07 


.19 




.14 




.22 




.22 




.21 




.23 








.33 


— .02 


.07 


.25 




—.05 


.36 




.13 




.16 




.01 




.47 








.31 




.03 


! -14 




21 


.19 


— 


.14 




.06 




.08 




.30 




.22 




.31 






.06 


-.05 


.23 




—.05 


.32 




.02 




.iF 




—.03 




.31 




.33 








.03 


\ —.01 




.06 


—.26 




.06 




-.05 




.25 





.13 





.02 




.03 




.03 




(.05 


.06 




.05 


—.15 




—.2? 




—.13 




.11 




—.17 




—.02 




.06 







52 Tests of Mathematical Ability and Their Prognostic Value 



TABLE VI 



Crude Coefficients— Horace Mann School 



B B 



Algebraic Computation 

Algebraic Computation 

Matching Equations and Problems. 
Matching Equations and Problems. 
Matching Nth Terms and Series. . 
Matching Nth Terms and Series.. 

Interpolation 

Interpolation 

Missing Steps in Series 

Missing Steps in Series 

Inference with Symbols 

Inference with Symbols 

Geometry 

Geometry 

Superposition 

Superposition 

Symmetry 

Symmetry 

Matching Solids and Surfaces 

Matching Solids and Surfaces 

Geometrical Definitions 

Geometrical Definitions 

Reasoning 

Reasoning 

Arithmetic Problems 

Arithmetic Problems 

Mixed Relations 

Mixed Relations 

Logical Opposites 

Logical Opposites 

Trabue Language Scales 

Trabue Language Scales 

Thorndike Reading Tests 

Thorndike Reading Tests 

Age 

Age 









5 S 


*n 'u 
















o o 


V V 
















0^ P4 


T3 -d 
















-O t3 


c c 












a 


c 


ti a 


rt ti 




?? ? 


.2 ^ 






o 


o 


ti d 






« « 


o o 






rt 


1 


CO en 

II 

;3 3 






.s .s 


W CO 

A A 






U 


6 


W W 


% % 


1 -2 


f 1 


1 1 






o 


o 


&o bo 


bo bo 


td rt 


W C/3 


U V 


*1 V 




1 


1 


.S .S 


.S .S 


1 1 


bo bo 

.s .s 


o o 

a a 




t 


t 






« a> 


"5) *OT 

.2 .2 


^ ^ 


B 8 
o o 




< 


< 


^ ^ 


1^ S 




% % 


1-4 1— ! 


O 6 






(St 


»H (M 


»-« <M 


^ <M 


rH OJ 


!-< M 


»H <M 


1 






.66 


.45 


.50 


.58 


.50 


.47 ; 


2 






.59 


.32 


.65 


.62 


.43 


.52 


1 




.59 




.40 


.32 


.55 


.28 


.48 


2 


.66 






.21 


.41 


.52 


.53 


.38 


1 




.32 


.21 




.30 


.31 


.15 


.18 


2 


.45 




.40 




.38 


.45 


.38 


.29 


1 




.65 


.41 


.38 




.63 


.49 


.53 


2 


.50 




.32 


.30 




.73 


.45 


.45 ! 


1 




.62 


.52 


.45 


.11 




.48 


.42 1 


2 


.58 




.55 


.31 


.63 




.40 


.42 I 


1 




.43 


.53 


.38 


.45 


.40 




.43 .1 


2 


.50 




.28 


.15 


.49 


.48 




.28 


1 




.52 


.38 


.29 


.45 


.42 


.28 


I 


2 


.47 




.48 


.18 


.53 


.42 


.43 




1 




.51 


.34 


.17 


.31 


.36 


.28 


.48 ' 


2 


.34 




.34 


.04 


.43 


.25 


.32 


.44 ; 


1 




.28 


.16 


.10 


.24 


.40 


.10 


.35 • 


2 


.21 




.32 


.17 


.32 


.23 


.07 


.32 


1 




.26 


.09 


—.04 


.25 


.23 


.19 


.38 


2 


.22 




.09 


—.00 


.27 


.24 


.19 


.52 


1 




.47 


.33 


.15 


.35 


.41 


.32 


.58 


2 


.03 




.31 


.01 


.31 


.39 


.38 


.57 


1 




.36 


.20 


.24 


.15 


.21 


.32 


.45 ji 


2 


.30 




.30 


.14 


.29 


.35 


.40 


.42 


1 




.63 


.54 


.36 


.45 


.55 


.45 


.32 


2 


.41 




.28 


.17 


.51 


.48 


.25 


.33 


1 




.36 


.23 


.11 


.36 


.48 


.19 


.41 


2 


.33 




.27 


.05 


.32 


.32 


.19 


.35 


1 




.47 


.44 


.21 


.31 


.41 


.24 


.32 


2 


.32 




.36 


.14 


.32 


.29 


.20 


.28 


1 




.12 


—.04 


.17 


.06 


.27 


—.12 


.22 


2 


.26 




.30 


.19 


.25 


.24 


.25 


.32 


1 




.34 


.32 


.06 


.26 


.24 


.24 


.29 i 


2 


.43 




.42 


.29 


.37 


.41 


.23 


.45 


1' 




.41 


—.29 


—.25 


—.52 


—.43 


—.40 


—.40 


2 


—.46 




—.29 


—.13 


-.43 


—.37 


—.52 


—.48 



The Analysis of Mathematical Ability 



53 



Table VI — Continued 







































































3 


3 




m 














C« 


W 


G 


(3 














•o 


"O 


O 


_o 














c 


c 


+J 


.♦J 














rt 


m 


c 


a 






§ 


§ 








CO 

C/3 


Q 


Q 
13 






*« 


'ot 


t 


b 


bo 


bo 


.o 


3 


^ 


to 


1 


£• 


it 

S 


8 


c 

1 




v 


is 


a 
o 






c^ 


s 


1 


s 


c4 


U 


o 




(4 
P< 



a a 



dH dH 



a a 



<: <3 



c c 
o o 



pe) P< 



S 1^ 



CI. a 
O O 



bo bo 
o o 

1-1 l-l 






H H 



bo bo 

S .S 

'•B "B 

a <i 

(^ Pi 



V V 

to bo 
< < 



.34 
.34 



.04 
.43 



.25 
.32 



.44 



.48 



.63 
.58 

.31 
.38 

.41 
.35 

.40 
.35 

.36 
.39 

.32 
.14 

.14 
.15 

.10 
.20 

.01 
.28 

—.20 
.17 



.17 



.10 — 



.63 



.58 



.28 
.39 

.31' 
.24 

.02 
.24 

.29 
.31 

.34 
.20 

.10 
.63 

.36 
.12 

.11 
.31 

—.09 
.14 



.54 
.43 

.45 
.51 

.39 
.12 

.30 
.31 

.24 
.18 

.22 
.22 

.17 
.22 

—.10 
.11 



.03 
.31 
.01 
.31 
.39 
.38 
.57 
.35 
.24 
.43 



.34 
.35 

.38 
.29 

.42 
.39 

.20 
.19 

.19 
.42 

.34 
.52 

—.39 
.28 



.36 

.20 



.45 
.40 



.35 
.40 



.24 
.51 



.35 



.34 



.40 
.34 



.54 



—.18 



.37 
-.23 



.41 

.28 
.17 
.51 
.48 
.25 
.33 
.39 
.31 
.12 
.29 
.21 



.33 

.44 

.32 
.24 

.15 
.46 

.18 
.47 

—.28 
.26 



.33 

.27 



.05 
.32 



.32 
.19 



.41 
.32 



.35 
,14 



.42 

.34 



.33 



.44 



.19 

.28 



.38 



.40 
.18 



.47 



.32 
.36 



.14 
.32 



.41 



.29 
.20 



.28 
.15 



.63 
.18 



.24 
.33 



.22 



.12 



.44 —.04 



.19 

.25 



.27 



.24 
.25 



.24 —.12 



.32 

.20 



.19 
.20 



.42 
.24 



.46 

.25 



.14 



.28 



—.20 —.19 — . 



.36 
49 

—.32 
11 



.43 

.34 — , 

.42 

.32 — . 



.06 



.29 
.37 



.26 — . 

.41 

.24 — , 

.33 

.24 — . 



.45 
.28 



.31 
.11 — 



.17 



.34 — 



.54 
.47 



.40 
.30 



.38 
.31 



.49 
.36 — , 



—.27 



.45 



—.46 
41 

—.29 
29 

—.13 
25 

—.43 
52 

—.37 
43 

—.52 
40 

—.48 
40 

—.17 
20 

—.14 
09 

—.11 
10 

—.28 
39 

—.18 
23 

—.26 
28 

—.20 
18 

—.19 
22 

—.11 
32 

—.27 
45 



54 Tests of Mathematical Ability and Their Prognostic Value 

TABLE VII 
Average Crude Coefficients — Wadleigh High School 





o 


V 










d 


§1 


H 






s 






crw 






a 






rt 


W3 


Za 






^« 




|l 


.si 


M£ 


1 

j3 


03 « 
bog 


^ 




li 


^1 


1^ 


(0 fl 




s 
s 


^ 


^ 


1^ 


t-i 


§ 


M 


o 




.46 


[.12] 


.36 


.55 




.28 


.46 




[.17] 


[.14] 


.29 


[.01] 


.21 


1.12J 


[.17: 




[.10] 


[.12] 


[.13] 


[.11] 


.36 


[.14 


[.10] 




.33 


.23 


.23 


.55 


.29 


[.12] 


.33 




[—.02] 


[.07] 


-.041 


[.01] 


.13] 


.23 


[—.02] 




.18 


.28 


.21 


.11] 


.23 


[.07] 


.18 




.27 


.21 


.19 


.26 


.20 


[.14] 


[.06] 


1.031 


.18 


[.10] 


[.09] 


[.05] 


.18 


.22 


.26 


[.10] 


i.l2] 


[.16] 


.22 


[.05] 


.23 


.21 


[.11] 


.09] 


[.12] 


[.13] 


[.15] 


.37 


.26 


.22 


[—.01] 


.35 


.21 


.18 


[.091 


.36 


.21 


[—.09] 


.32 


.23 


[.12] 


[.17 1 


l.ioj 


.32 


[—.13] 


[.16] 


[.10] 


[.13] 


.23 


.28 


.29 


[.12] 


.24 


.29 


.28 


.21 


.32 


.31 


[.13] 


[.12] 


[.13] 


[.16] 


.29 


1.021 


.26 


[.09] 


.18 


[-.11] 


.22 


[.17] 


-.25 


[—.15] 


[—.08] 


—.37 


—.18 


[—.11] 


[—14] 



Algebraic Computation 

Matching Equations and Problems. 
Matching Nth Terms and Series... 

Interpolation 

Missing Steps in Series 

Inference with Symbols 

Geometry 

Superposition 

Symmetry 

Matching Solids and Surfaces 

Geometrical Definitions 

Reasoning 

Arithmetic Problems 

Mixed Relations 

Logical Opposites 

Trabue Language Scales 

Thorndike Reading Tests 

Age 



Coefficients of less than 2 P.E. are put in square brackets. 



TABLE VIII 
Average Crude Coefficients — Horace Mann School 





■5 «" 


If 
H 










s 


^i 


^ 


c 


U3 


X 






^^ 




o 


o. 


.ti 




a 


W3 


^ a 




^^ 


^<0 




.Si 3 


t«S 


MS 




t/3_4) 


r.^ 


C 


Hi 


.SPh 


.Sw 




Sffe 


fll 




•o 5 
Sc3 






£3 






t 


< 


^ 


S 


t— 1 


^ 


HH 


O 




.63 


,39 


.57 


.60 


.46 


.49 


.63 




.30 


.37 


.53 


.41 


.43 


.39 


.30 




.34 


.38 


.26 


.24 


.57 


.37 


.34 




.68 


.47 


.49 


.60 


.53 


.38 


.68 




.44 


.42 


.46 


.41 


.26 


.47 


.44 




.36 


.49 


.43 


.24 


.49 


.42 


.36 




.42 


.34 


[.11] 


.37 


.31 


.30 


.46 


.25 


.24 


[.14] 


.28 


.31 


[.08] 


.33 


.24 


[.09] 


[—.02] 


.26 


.23 


.19 


.45 


.25 


.32 


[.08] 


.33 


.40 


.35 


.57 


.33 


.25 


.19 


.22 


.28 


.36 


.43 


.52 


.41 


.27 


.48 


.52 


.34 


.33 


.34 


.25 


[.08] 


.34 


.40 


.19 


.38 


.39 


.19 


.28 


.31 


.35 


.22 


.30 


.19 


[.13] 


.18 


[.15] 


.26 


[.07] 


.27 


.39 


.37 


.18 


.32 


.32 


.28 


.37 


-.44 


—.30 


—.19 


—.48 


—.40 


—.46 


—.44 



Algebraic Computation 

Matching Equations and Problems 
Matching Nth Terms and Series.. 

Interpolation 

Missing Steps in Series 

Inference with Symbols 

Geometry 

Superposition 

Symmetry 

Matching Solids and Surfaces.... 

Geometrical Definitions 

Reasoning 

Arithmetic Problems 

Mixed Relations 

Logical Opposites 

Trabue Language Scales 

Thorndike Reading Tests 

Age 



Coefficients of less than 2 P.E. are put in square brackets. 



The Analysis of Mathematical Ability 
Table VII — Continued 



55 













1 


0) 


s 


j> 


.S 








«3 









a 


•3 


S? 


•a 




1 


>, 


-4 


CO 

So 


to 





•1 


1 




1 

rS 






s 


i4 


^S 


•c5 


.S 


4> 


fii 




►J CO 


^ 




1 

eg 


1 

a 


rt CO 


1^ 


o 


1 

"•§ 


1 






Is 


bo 

< 


.27 


[.03] 


.26 


.21 


.26 


[.04] 


[.10] 


.28 


.32 


[.02] 


—.25 


.21 


.18 


[.10] 


[.11] 


.22 


.21 


.32 


.29 


.31 


.26 


[—.15] 


.19 


[.10] 


.14] 


[.09] 


[—.01] 


[—.09] 


[-.13] 


[.12] 


[.13] 


[.09] 


[—.08] 


.26 


[.09] 


[.16] 


[.12] 


.35 


.32 


[.16] 


.24 


[.12] 


.18 


—.37 


.20 


[.05] 


.22 


[.13] 


.21 


.23 


[.10] 


.29 


[.13] 


[—.11] 


—.18 


[.14] 


.18 


[.05] 


[.15] 


.18 


[.12] 


[.13] 


.28 


[.16] 


.22 


[—.11] 


[.06] 


.22 


.23 


.37 


[.09] 


[.17] 


.23 


.21 


.29 


[.17] 


[—.14] 




.60 


.36 


.20 


.28 


.19 


[.17] 


[.08] 


[.03] 


[.05] 


[—.03] 


.60 




.24 


.37 


[.14] 


.27 


[.14] 


[.08: 


[.16] 


.22 


[.06] 


.36 


.24 




.19 


.24 


[.09] 


[.171 


[.12] 


[.07] 


[.07] 


[-.10] 


.20 


.37 


.19 




[.01] 


[—.02] 


[.14: 


.26 


.25 


[.09] 


[—.05] 


.28 


[.14] 


.24 


[.01] 




.22 


[.14] 


.20 


[.17] 


[.02] 


[—.13] 


.19 


.27 


[.09] 


[—.02] 


.22 




[—.05] 


[.06] 


.19 


[.02] 


—.19 


[.17] 


[.14] 


:.17] 


[.14] 


[.14] 


[—.05] 




.39 


[.11] 


[.14] 


[—.01] 


[.08] 


[.08] 


.12] 


.26 


.20 


[.06] 


.39 




.35 


.27 


[—.10] 


[.03] 


[.16] 


.07] 


.25 


[.17] 


.19 


[.11] 


.35 




.32 


[.00] 


[.05] 


.22 


.07] 


[.09] 


[.02; 


[.02] 


[.14] 


.27 


.32 




[.04] 


[—.03] 


[.06] 


[—.10] 


[—.05] 


[—.13] 


—.19 


[—.01] 


[—.10] 


[.00] 


[.04] 





Table VIII — Continued 













S 


«» 


1 


«> 


bo 

s 




§ 


1 


CO 

Gin 


s.i 


.S 


^ 


1 




1 


bo 

1 


1 




0, 


H 


ii a 


C4 




X 


bo 


-5 S 


Or« 


« 


^ 




CTl OS 


2« 



rt 


< 


ii 


^ 






bo 

< 


.42 


.25 


.24 


.25 


.33 


.52 


.34 


.39 


.19 


.39 


— .44 


.34 


.24 


[.09] 


.32 


.25 


.41 


.25 


.19 


[.13] 


.37 


—.30 


[.11] 


[.14] 


[—.02] 


[.08] 


.19 


.27 


[.08] 


.28 


.18 


.18 


—.19 


.37 


.28 


.26 


.33 


.22 


.48 


.34 


.31 


[.15] 


.32 


—.48 


.31 


.31 


.23 


.40 


.28 


.52 


.40 


.35 


.26 


.32 


—.40 


.30 


[.08] 


.19 


.35 


.36 


.34 


.19 


.22 


[.07] 


.28 


—.46 


.46 


.33 


.45 


.57 


.43 


.33 


.38 


.30 


.27 


.37 


—.44 




.61 


.35 


.38 


.37 


.37 


.23 


[.14] 


[.15] 


[.14] 


—.18 


.61 




.33 


.28 


.28 


.29 


.27 


.36 


.24 


.21 


[—.12] 


.35 


.33 




.49 


.48 


.25 


.31 


.21 


.22 


.20 


[—.11] 


.38 


.28 


.49 




.35 


.34 


.40 


.19 


.31 


.43 


—.33 


.37 


.28 


.48 


.35 




.31 


.27 


.26 


.22 


.45 


—.21 


.37 


.29 


.25 


.34 


.31 




.39 


.28 


.31 


.32 


—.27 


.23 


.27 


.31 


.40 


.27 


.39 




.26 


,26 


.39 


—.19 


[.14] 


.36 


.21 


.19 


.26 


.28 


.26 




.21 


.31 


—.21 


[.15] 


.24 


.22 


.31 


.22 


.31 


.26 


.21 




.42 


—.22 


[.14] 


.21 


.20 


.43 


.45 


.32 


.39 


.31 


.42 




—.36 


—.18 


[-.12] 


[-.11] 


—.33 


—.21 


—.27 


—.19 


—.21 


—.22 


—.36 





56 Tests of Mathematical Ability and Their Prognostic Value 

TABLE IX 
Crude Coefficients — Wadleigh High School and Horace Mann School 





0» 

s 




g 












"ti en 


H 










g 


sE 


X 


g 


eo 


^ 




.S 3 




00 i; 


rt 






^ 


II 


.Sf^ 


.S^ 


0. 


bfS 


fli 


■5 


n 


11 

at rt 


£ 




P 


1 


<; 


S 


^ 


M 


s 


M 






.57 


.30 


.50 


.59 


.29 


.42 


.57 




.26 


.29 


.45 


.25 


.36 


.30 


.26 




.26 


.29 


.22 


.19 


.50 


.29 


.26 




.57 


.39 


.40 


.59 


.45 


.29 


.57 




.29 


.31 


.29 


.25 


.22 


.39 


.29 




.30 


.42 


.36 


.19 


.40 


.31 


.30 




.37 


.29 


.13 


.33 


.27 


.24 


.32 


.17 


.22 


.12 


.22 


.22 


.12 


.29 


.25 


.09 


.03 


.23 


.22 


.14 


.37 


.24 


.25 


.08 


.26 


.31 


.29 


.50 


.31 


.24 


.12 


.26 


.25 


.30 


.32 


.36 


.34 


.15 


.43 


.42 


.27 


.27 


.26 


.27 


.01 


.28 


.30 


.17 


.33 


.36 


.23 


.22 


.29 


.33 


.24 


.27 


.23 


.19 


.16 


.14 


.21 


.10 


.28 


.26 


.33 


.14 


.27 


.18 


.26 


.30 


-.37 


—.25 


—.15 


—.44 


—.33 


—.34 


—.34 



Algebraic Computation 

Matching Equations and Problems.... 

Matching Nth Terms and Series 

Interpolation 

Missing Steps in Series 

Inference with Symbols 

Geometry 

Superposition 

Symmetry 

Matching Solids and Surfaces 

Geometrical Definitions 

Reasoning 

Arithmetic Problems 

Mixed Relations 

Logical Opposites 

Trabue Language Scales 

Thorndike Reading Tests 

Age 



These crude coefficients, however, do not tell us accurately the 
real amount of correlation that exists in the case of any two func- 
tions. Apart altogether from the error due to sampling, of whose 
size we can judge and in the case of which we can protect our- 
selves from false reasoning by estimating the probable error, there 
are other important sources of fallacy. Thus not only do our 
crude coefficients represent the correlations found in a very limited 
group of individuals, they are also merely such measures of cor- 
respondence as arise from two sets of observations obtained by 
methods of experimentation more or less imperfect. Errors of 
the latter kind, which can assume large proportions in psy- 
chological work, cannot be got rid of by increasing the number 
of individuals examined. They do not tend to balance each other 
in the case of correlations as happens in determining group 
averages. Their tendency is to reduce the size of the coefficients 
calculated towards zero. In order to eliminate their effect, Spear- 
man * has proposed certain formulae, "based on the idea that the 



* Spearman, C, The Proof and Measurement of Association Between 
Two Things, Am. Jour. Psych. XV: 88, and Correlation calculated from 
Faulty Data, British Jour. Psych. Ill : 271. 



The Analysis of Mathematical Ability 57 

Table IX — Continued 



, 


>» 


wjS 


1 

Q 
1 


tc 


2 

,0 


1 


8 

a 



c4 


f 




1 


v 

3 

3 


boh 

II 

1^ 




i 


< 


« 


*3 

1 


r 




1 


.37 


.17 


.25 


.24 


.31 


.04 


.26 


.36 


.23 


.26 


—.37 


.29 


.22 


.09 


.25 


.24 


.34 


.27 


.23 


.19 


.33 


—.25 


.13 


.12 


.03 


.08 


.12 


.15 


.01 


.22 


.16 


.14 


—.15 


.33 


.22 


.23 


.26 


.26 


.43 


.28 


.29 


.14 


.27 


—.44 


.27 


.22 


.22 


.31 


.25 


.42 


.30 


.33 


.21 


.18 


—.33 


.24 


.12 


.14 


.29 


.30 


.27 


.17 


.24 


.10 


.26 


—.34 


.32 


.29 


.37 


.50 


.32 


.27 


.33 


.27 


.28 


.30 


—.34 




.61 


.35 


.32 


.34 


.31 


.21 


.12 


.11 


.11 


—.13 


.61 




.30 


.31 


.23 


.29 


.22 


.27 


.21 


.21 


—.06 


.35 


.30 




.39 


.40 


.20 


.26 


.18 


.17 


.15 


—.09 


.32 


.31 


.39 




.23 


.22 


.31 


.31 


.28 


.32 


—.26 


.34 


.23 


.40 


.23 




.28 


.22 


.24 


.20 


.31 


—.18 


.31 


.29 


.20 


.22 


.28 




.23 


.23 


.33 


.23 


—.28 


.21* 


.22 


.26 


.31 


.22 


.23 




.43 


.28 


.35 


—.13 


.12 


.27 


.18 


.31 


.24 


.23 


.43 




.37 


.39 


—.17 


.11 


.21 


.17 


.28 


.20 


.33 


.28 


■ .37 




.50 


—.14 


.11 


.21 


.15 


.32 


.31 


.23 


.35 


.39 


.50 




—.23 



.13 —.06 —.09 —.26 —.18 —.28 —.13 —.17 —.14 —.23 

size of these accidental errors can be measured by the size of the 
discrepancies between successive measurements of the same 
things." These formulae have been criticised adversely by several 
writers,^ the most serious charge levelled against them being that 
their assumption that errors of observation are themselves un- 
correlated is unwarranted. Spearman admits the justice of this 
criticism in the case of "variations of a regular and continuously 
progressive character/' while insisting that there is besides these 
a host of "variations of a discontinuously shifting sort" that cannot 
be controlled, as the former may, which are due to accident, and 
which are most scientifically dealt with by a process of elimination 
comparable to the familiar methods of "smoothing curves" or 
"taking means." Udny Yule has demonstrated the existence of 
the attenuation of coefficients of correlation by errors of observa- 
tion by a still simpler proof and has shown the assumptions on 
which Spearman's Correction formulae are based. Spearman ^ 

^ Pearson, Karl, Biometrika, III : 160, and Drapers' Company Research 
Memoirs, Biometric Series, IV, 1907. Brown, Wm., The Essentials of 
Mental Measurement, Cambridge, 1911, 83. 

6 Spearman, C, General Intelligence — Objectively Determined and 
Measured, Am. Jour, Psych., XV: 257. 



58 Tests of Mathematical Ability and Their Prognostic Value 

has pointed out clearly the conditions under which correction can 
legitimately be applied. 

If the observed coefficient of correlation is less than twice the 
probable error, since there is no conclusive evidence of the ex- 
istence of positive correspondence between the traits under investi- 
gation, correction is out of the question. Where, however, the 
observed correlation is substantially greater than the probable 
error, say four or five times its amount, correspondence being es- 
tablished, we are justified in using a reasonable method of cor- 
rection in order to bring the attenuated measure nearer to its 
most probable true value. 

In the present investigation the particular formula ^ used was 
the following: 



V {^PlQ2) (^P2«l) 

rj)a — 



pq 



V y^PlPz) \'^QlQ2) 



Tpq here indicates the true correlation between two series of 
measures p and q of the facts A and B. 
P^ and />2 are two independent measures of A. 
q^ and ^2 ^^^ two independent measures of B, 

^PiQ2 ^^ ^^^ correlation obtained from the first measure of A and 

the second measure of B. 
Tpggj is the correlation obtained from the second measure of A and 

the first measure of B. 
^PiP2 ^^ *^^ correlation between the two measures of A. 
^flifl2 ^^ *^^ correlation between the two measures of B, 

In Tables X and XI the corrected coefficients for the two 
groups are given in full, even in cases where the low correlation 
and the high probable error scarcely warrant the correction being 
made. Where a coefficient is absent from the table it signifies that 
the Correction formula could not be applied owing to one of the 
crude coefficients being zero or the two being of unlike sign. In 
Table XII the results derived from both groups are amalgamated, 
those of Horace Mann School receiving double weight on account 
of their superior reliability. 

' Thorndike, E. L., Theory of Mental and Social Measurements, New 
York, 1913, 179. 



The Analysis of Mathematical Ability 59 

The probable error of these corrected coefficients may be ap- 
proximately determined by the use of the formula : ® 

P. E. of Corrected Coefficient Corrected Coefficient 



P. E. of Crude Coefficient Crude Coefficient 

Since it exceeds the probable error of the corresponding raw co- 
efficient in proportion to the amount of correction made, it is 
left to the reader to infer the required increase in each case. 

8 Burt, C, Experimental Tests of General Intelligence, British Jour. 
Psych. Ill: 111. 



6o Tests of Mathematical Ability and Their Prognostic Value 

TABLE X 
Corrected Coefficients — Wadleigh High School 



i 


§ 

la 


1 


C! 


(0 


jS 




o 


a'v 


.*' m 


5 


a 


,tJ 




u 

•s 


H3 






?5 






% 


II 


1^ 

ce ft 


V 

j3 


U3 C 


2J| 




< 


S 


S 


t-H 


1^ 


M 


a 




.65 


.15 


.48 


.72 




.37 


65 




.22 


.09 


.42 


.02 


.31 


.15 


.22 




.11 


.06 


.14 


.11 


48 


.09 


.11 




.45 


.54 


.25 


72 


.42 


.06 


.45 






.00 




.02 


.14 


.54 






.45 


.37 


.31 


.11 


.25 


.00 


.45 




.33 


.25 


.72 


.32 


.27 


.31 


.08 




.21 


, 


.11 


.05 


.38 


.29 


.46 


.08 


.25 


.30 


.43 


.03 


.34 


.38 


.15 


.11 




.27 




.81 


.44 


.33 




.63 


.36 


.61 


.17 


.05 


.33 


—.14 


.56 


.39 


.38 


.27 


.08 


.46 


—.25 


.23 


.15 


.33 


.40 


.40 


.49 




.33 


.42 


.72 


.32 


.54 


.64 


.18 


.12 


.21 


.51 


.52 




.44 


.06 


.30 


—.16 


.65 




.30 


—.18 


—.03 


—.44 


—.21 




—.12 


TABLE XI 













Algebraic Computation 

Matchinc^ Equations and Problems. . . 

Matching Nth Terms and Series 

Interpolation 

Missing Steps in Series 

Inference with Symbols 

Geometry 

Superposition 

Symmetry 

Matching Solids and Surfaces 

Geometrical Definitions 

Reasoning 

Arithmetic Problems 

Mixed Relations 

Logical Opposites 

Trabue Language Scales 

Thorndike Reading Tests 

Age 



Corrected Coefficients — Horace Mann School 



i 


ifi 


H 




CO 


J3 




CJ 


cri* 




a 


a 


.ti 




"S 

u 


W3 
beg 


.11 


o 
1 






«5 


bO 






o. 

u 
2i 






u 


< 


1^ 


^ 


*-< 


§ 


^ 


O 




.91 


.48 


.65 


.81 


.58 


.58 


.91 




.41 


.48 


.82 


.55 


.64 


.48 


.41 




.38 


.49 


.29 


.30 


.65 


.48 


.38 




.84 


.54 


.58 


.81 


.82 


.49 


.84 




.58 


.58 


.58 


.55 


.29 


.54 


.58 




.44 


.58 


.64 


.30 


.58 


.58 


.44 




.51 


.48 


.10 


.42 


.40 


.37 


.59 


.23 


.30 


.15 


.29 


.37 


.09 


.39 


.32 


.14 


—.01 


.32 


.33 


.25 


.62 


.14 


.45 


.04 


.37 


.52 


.43 


.72 


.43 


.37 


.24 


.25 


.38 


.57 


.59 


.72 


.64 


.35 


.63 


.79 


.47 


.48 


.43 


.36 


.09 


.39 


.51 


.24 


.49 


.57 


.67 


.25 


.42 


.55 


.32 


.45 


.34 




.35 


.22 


.53 




.53 


.57 


.62 


.19 


.42 


.49 


.41 


.55 


-.49 


—.38 


—.20 


—.49 


—.48 


—.51 


—.50 



Algebraic Computation 

Matching Equations and Problems 
Matching Nth Terms and Series. . 

Interpolation 

Missing Steps in Series 

Inference with Symbols 

Geometry 

Superposition 

Symmetry 

Matching Solids and Surfaces .... 

Geometrical Definitions 

Reasoning 

Arithmetic Problems 

Mixed Relations 

Logical Opposites 

Trabue Language Scales 

Thorndike Reading Tests 

Age 



The Analysis of Mathematical Ability 
Table X — Continued 



6i 













5 


(0 


-»j 


M 


s 




G 




Is 






2 


1 




03 


^ 

s 




_o 


>, 




si 


bo 


_o 




o 


C 


S CO 




a 


1 

a 


1^ 




c 
'c 
o 

a 


s 


.2 


o 

'So 




(U 


p 


>> 


rt nJ 


SQ 


« 


'C 




o 


[«C/2 




bo 


w 


c^ 


1^ 


O 


ce; 


< 


ji 


h-1 


H 


< 


.33 




.46 


.38 


.44 


.05 


.08 


.40 


.54 




—.30 


.25 


.21 


.08 


.15 


.33 


.33 


.46 


.49 


.64 


.44 


—.18 


.72 




.25 


.11 




—.14 


—.25 




.18 


.06 


—.03 


.32 


.11 


.30 




.63 


.56 


.23 


.33 


.12 


.30 


—.44 


.27 


.05 


.43 


.27 


.36 


.39 


.15 


.42 


.21 


—.16 


—.21 


.31 


.38 


.03 




.61 


.38 


.33 


.72 


.51 


.65 




.08 


.29 


.34 


.81 


.17 


.27 


.40 


.32 


.52 




—.12 




.68 


.67 


.37 


.47 


.26 


.28 


.03 






—.03 


.68 




.27 


.69 


.19 


.41 


.09 




.20 


.33 


.06 


.67 


.27 




.06 


.59 




.30 


.06 








.37 


.69 


.06 










.58 


.60 






.47 


.19 


.59 






.*47 


132 


.37 


.38 




—.15 


.26 


.41 






.47 




—.06 




.41 




—.26 


.28 


.09 


".30 




.32 


—.06 




.74 


.07 






.03 




.06 


.'58 


.37 




.74 




.60 


.47 


—.08 




120 


* 


.60 


.38 


!41 


.07 


.60 




.67 






.33 












.47 


.67 




.06 


— .'03 


.06 






—.15 


— !26 




—.08 




.'06 





Table XI — Continued 



a 
o 




3 <« 








C 

o 


a 












cojS 


-^ c 






01 


o. 


G 


Ph 




.ti 


>t 




3.2 


be 


u 


'v 


o 


0! 






in 


is 


bos 




c 


ai 


« 




^« 


,i< 




o. 


V 


CU3 


II 


"c 
2 








3 rt 






Cu 


p 


Ba 


S V 


rt 


.-^ 


X 


% 


XI 3 


^r*^ 


V 


^ 


^ 








< 


ji 


o 




r 


bO 

< 


.51 


.23 


32 


.14 


.43 


.72 


.43 


.57 


.34 


.57 


—.49 


.48 


.30 


.14 


.45 


.37 


.64 


.36 


.67 




.62 


—.38 


.10 


.15 


—.01 


.04 


.24 


.35 


.09 


.25 


.35 


.19 


—.20 


.42 


.29 


.32 


.37 


.25 


.63 


.39 


.42 


.22 


.42 


—.49 


.40 


.37 


.33 


.52 


.38 


.79 


.51 


.55 


.53 


.49 


—.48 


.37 


.09 


.25 


.43 


.57 


.47 


.24 


.32 




.41 


—.51 


.59 


.39 


.62 


.72 


.59 


.48 


.49 


.45 


.53 


.55 


—.50 




.68 


.46 


.46 


.48 


.53 


.26 


.21 


.28 


.06 


—.20 


68 




.40 


.31 


.08 


.38 


.30 


.34 


.36 


.25 


—.12 


.46 


.40 




.63 


.67 


.33 


.45 


.32 


.46 


.31 


—.13 


.46 


.31 


.63 




.44 


.46 


.49 


.28 


.54 


.61 


—.36 


.48 


.08 


.67 


.44 




.44 


.34 


.36 


.45 


.69 


—.24 


.53 


.38 


.33 


.46 


.44 




.55 


.47 


.59 


.49 


—.41 


.26 


.30 


.45 


.49 


.34 


.55 




.37 


.52 


.58 


—.22 


.21 


.34 


.32 


.28 


.36 


.47 


.37 




.45 


.54 


—.27 


.28 


.36 


.46 


.54 


.45 


.59 


.52 


.45 




.96 


—.33 


.06 


.25 


.31 


.61 


.69 


.49 


.58 


.54 


.96 




—.46 


—.20 


—.12 


—.13 


—.36 


—.24 


—.41 


—.22 


—.27 


—.33 


—.46 





62 Tests of Mathematical Ability and Their Prognostic Value 

TABLE XII 

G)RRECTED Coefficients — Wadleigh High School and 
Horace Mann School 







1 
















Pi 


•d 












1 


1 

en 

a 


§ 

i 




1 


•i 
a 






■♦* 


o 


« 






>« 






1 


at 


H 




.5 


w 








s 


^ 




m 


A 






c3 


S 


^ 


1 


s 


'% 






,o 


\ili 


bo 


ta 


xn 


0) 


>. 




*C3 

1 


1 


a 
1 


o 


bo 

c 


1 


1 




< 


:^ 


§ 


c 


i 







Algebraic Computation 




.82 


.37 


.60 


.78 


.35 


.51 


Matching Equations and Problems.... 


.82 




.35 


.35 


.68 


.37 


.53 


Matching Nth Terms and Series 


.37 


.35 




.29 


.35 


.24 


.23 


Interpolation 


.60 


.35 


.29 




.71 


.54 


.47 


Missing Steps in Series 


.78 


.68 


.35 


.71 




.39 


.39 


Inference with Symbols 


.35 


.37 


.24 


.54 


.39 




.45 


Geometry 


.51 


.53 


.23 


.47 


.39 


.45 




Superposition 


.45 


.40 


.31 


.39 


.35 


.35 


.42 


Symmetry 


19 


27 


10 


.23 


.26 


19 


35 


Matching Solids and Surfaces 


.37 


.12 


.09 


!31 


!36 


.'I8 


!53 


Geometrical Definitions 


.22 


.35 


.06 


.31 


.44 


.29 


.75 


Reasoning 


.44 


.35 


.16 


.38 


.37 


.58 


.45 


Arithmetic Problems 


.50 


.53 


.19 


.61 


.66 


.44 


.41 


Mixed Relations 


.32 
.51 


.39 
.61 


—.02 
.17 


.34 
39 


.39 
50 


.27 
46 


.46 


Logical Opposites 


41 


Trabue Language Scales 


.41 


.70 


.29 


.18 


.42 


.40 


!53 


Thorndike Reading Tests 


38 


56 


15 


.38 


.33 


.49 


yj 


Age 


— !43 


—.31 


—.15 


-.47 


—.'39 


—.'34 


— !38 



The Analysis of Mathematical Ability 63 

Table XII — Continued 







1 


.2 

"3 




i 




M 













o» 


(A 




3 


_ 


a 


2? 


'•B 




G 



1 


12 

1 
1 

u 


4) 

Q 
g 


to 
.S 

a 

g 




u 




•J 


s 

0. 



.1 


3 
M 

C 

1 


cm 

s 




0. 


c 







rt 


,-^ 


>< 


bo 


rt 





V 


3 




<0 


V 


4J 


'u 







2 


.c 


bo 


03 


w 


s 





(^ 


< 


i 


_! 


H 


H 


< 


.45 


.19 


.37 


.22 


.44 


.50 


.32 


.51 


.41 


.38 


—A3 


.40 


.27 


.12 


.35 


.35 


.53 


.39 


.61 


.70 


.56 


-.31' 


.31 


.10 


.09 


.06 


.16 


.19 


—.02 


.17 


.29 


.15 


—.15 


.39 


23 


.31 


.31 


.38 


.61 


.34 


.39 


.18 


.38 


—.47 


.35 


.26 


.36 


.44 


.37 


.66 


.39 


.50 


.42 


.33 


—.39 


.35 


.19 


.18 


.29 


.58 


.44 


.27 


.46 


.40 


.49 


—.34 


.42 


.35 


.53 


.75 


.45 


.41 


.46 


.41 


.53 


.37 


—.38 




.68 


.53 


.43 


.48 


.44 


.27 


.15 


.19 


.04 


—.14 


.68 




.36 


.44 


.12 


.39 


.23 


.23 


.31 


.28 


—.10 


.53 


.36 




.44 


.64 


.22 


.40 


.24 


.50 


.21 


—.09 


.43 


.44 


.44 




.29 


.31 


.33 


.38 


.56 


.41 


—.24 


.48 


.12 


.64 


.29 




.45 


.33 


.36 


.42 


.46 


—.21' 


.44 


.39 


.22 


.31' 


.45 




.35 


.31 


.53 


.33 


—.32 


.27 


.23 


.40 


.33 


.33 


.35 




.49 


.37 


.45 


—.14 


.15 


.23 


.24 


.38 


.36 


.31 


.49 




.50 


.51 


—.21 


.18 


.31 


.50 


.56 


.42 


.53 


.37 


.50 




.87 


—.22 


.04 


.28 


.21 


.41 


.46 


.33 


.39 


.51 


'.S7 




—.29 


—.14 


—.10 


—.09 


—.24 


—.21 


—.32 


—.14 


—.21 


—.22 


—.29 





64 Tests of Mathematical Ability and Their Prognostic Value 

We can now turn these statistical results to account in analyz- 
ing the degree and the kind of interdependence that exists be- 
tween the various mental functions measured. Certain general 
features characterizing all the Correlation tables are worthy of 
note. In spite of the complexity of the interrelations which they 
show, theoretical conclusions of value can be deduced from a 
rapid survey. There is apparent a tendency for allied tests to 
correlate together more closely than those from different groups. 
Thus coefficients derived from pairs of tests of algebraic abilities 
are in general higher than those that ensue from combining two 
tests, one of algebraic and the other of geometrical capacities. 
An analysis of the tables yields the following interesting re- 
sults. 

In Table VII ten of the coefficients of correlation between 
mathematical functions exceed .50 and of these every single pair 
of tests belong to the same group of abilities. In the same table 
.26 of the coefficients of correlation between mathematical ca- 
pacities are greater than .40 and of these twenty-one are derived 
from tests of the same class. Similarly, in Table VIII, of 26 co- 
efficients exceeding ,40 in amount, twenty-one are obtained from 
tests of allied abilities. 

The results are as marked in the case of the corrected coef- 
ficients. In Table X, of eleven coefficients over .50, ten issue 
from pairs of functions both of which belong either to the group 
of algebraic abilities or to that of geometrical abilities, and 
similarly in Table XI, of twenty-three coefficients exceeding .55, 
nineteen result from tests of the same general kind. 

The combination of the Wadleigh and the Horace Mann coef- 
ficients, both raw and corrected, in Tables IX and XII, offers 
equally striking evidence. 

This tendency can be generally discerned in the changes in the 
magnitude of the correlations, as we pass from the tests belong- 
ing to one field to those of another. It is perhaps most obvious 
in the case of the somewhat specialized tests involving intuitive 
grasp of spatial relations. 

Another feature common to all the Correlation tables is the 
absence of negative coefficients with the one exception of the al- 
most universal negative correlation of every function tested with 



The Analysis of Mathematical Ability 65 

age. Apart from the latter, each negative coefficient found is 
less than twice its probable error. Consequently it may be due 
to accidental flaws in the method of measurement and cannot be 
regarded as demonstrating the presence of inverse relation. 
Practically all such are cases of absence of correspondence be- 
tween the traits measured. Even for tasks apparently so dis- 
similar as the solving of arithmetic problems, the interpolation of 
numbers and the superposition of geometrical figures, a positive 
correspondence exists, though frequently it is small in amount. 

There is an obvious tendency for age to correlate inversely with 
the functions measured. Thus it is apparent that the tests are 
indicative of the qualities which cause a pupil to begin the study 
of mathematics young, or to progress through school rapidly, 
or both. 

The Correlation tables give a partial analysis of mathematical 
intelligence, expressing in precise quantitative form the kind and 
amount of kinship between the various abilities examined. For 
deeper insight into the nature of these traits further statistical 
treatment is necessary. Up to this point we have considered 
them in isolation. We must now pass to the inquiry into the rel- 
ative status of each in mathematical intelligence. At the same 
time we shall also determine the characteristics in a test which 
produce high correlation with mathematical ability, and discover, 
if possible, the common psychological factor or factors which ex- 
plain the manifold correspondences that appear in the tables. 

For this purpose we require a measure of mathematical ability 
with reference to which we can determine the value of each test. 
We might obtain such a measure in a variety of ways. It is de- 
sirable, for example, that we should have an independent estimate 
of the efficiency in algebra, geometry, and arithmetic of the pupils 
examined. This was in fact obtained from the school marks in 
these subjects, which they had received up to date. 

A second measure that might be used as an index to general 
proficiency in mathematical work is the grand total of the scores 
in the tests of all of the functions which can lay substantial 
claim to be called mathematical. 

A third possible method of determining the relative value of 
the tests as measures of mathematical ability is by comparison of 



66 Tests of Mathematical Ability and Their Prognostic Value 

the average of any particular test's correlations with all the others 
with their corresponding averages.^ 

Each of these standards was used in this study. In the case 
of the first and third, the procedure is simple and straightforward, 
requiring no explanation. It is necessary, however, to give some 
account of how our second standard was derived. 

The difficulties in the way of combining the results of several 
tests are the incommensurability of certain measures (some be- 
ing in terms of time and others in terms of accuracy), the dif- 
ferent averages of the same group in different tests, and the dif- 
ferent variabilities of the same group in different tests. To 
eliminate both the absolute value of the average and that of the 
variability Woodworth ^^ has suggested that we let the average 
be counted as zero, so that the standing of each individual is 
expressed as a deviation, and to make the measure of variability 
the unit deviation, so that all deviations are expressed as fractions 
or multiples of this unit. Thus each individual in each test is 
assigned a position in the distribution of the group. He stands 
above or below the group average and so much above or below as 
compared with the average variation of the group. Thus hav- 
ing determined each individual's position with reference to the 
central tendency in the case of each test, these values are reduced 
to suitable proportions to one another. If we desire all the tests 
combined to have an equal influence in the composite, we must 
multiply the deviations by such factors as will make their varia- 
bilities equal. Where, however, we wish for any reason to attach 
greater weight to certain tests than others we must multiply the 
deviations of the tests in question by such factors as will make 
their variability greater in the required proportion. 

It is certainly desirable in constructing a composite measure 
of the individual's achievement in the several tests that we should 
take cognizance of the factors exercising a significant influence 
upon the relations to be investigated. Obviously these may be 

® Compare McCall, W. A., Correlation of Some Psychological and Edu- 
cational Measurements, Columbia University, Contributions to Education, 
Teachers College Series, No. 79, 35. 

10 Woodworth, R. S., Combining the Results of Several Tests, Psycho- 
logical Review, XIX: 97 and Yule, G. Udny, An Introduction to the 
Theory of Statistics, 218-219, 1112, Correlation due to Heterogeneity of 
Material, London, 1916. 



The Analysis of Mathematical Ability 6y 

numerous, but some we shall have to neglect because we lack 
the knowledge necessary to decide what importance to attach to 
them. The tests certainly differ in their value as indices to math- 
ematical ability and this would have to be recognized in a per- 
fect measure of general mathematical efficiency. The best weights 
to attach to each test might be determined by the method of partial 
correlation coefficients, which has been devised by Edgeworth,^^ 
Pearson ^^ and Yule ^^ and developed by Kelley.^* The method, 
however, becomes exceedingly cumbrous and the labor it involves 
enormous, where the variables are at all numerous, and probably 
sufficiently satisfactory results can be had, where almost any 
reasonable weighting is used. 

The latter empirical method of arriving at a best possible 
composite for mathematical ability was the one followed in this 
study. The measures for each function were weighted with 
reference to two main factors, the importance of the ability 
measured and the reliability of the test from which they were de- 
rived. In accordance with these principles six of the tests were 
given double weight in the composite developed. These were 
Algebraic Computation, Matching Equations and Problems, Ge- 
ometry Test, Matching Solids and Surfaces, Interpolation, and 
Arithmetic Problems. It will be seen that these include the most 
reliable tests and, as far as we can judge, the most representative 
tests of the total number. 

In Tables XIII and XIV are given the data upon which the new 
measures of mathematical ability compounded from all the tests 
were based. 

11 Edgeworth, F. Y., On Correlated Averages, Phil Mag. 5th Series, 
XXXIV: 194. 

12 Pearson, Karl, Regression, Heredity, and Panmixia, Phil. Trans. 
Roy. Soc, Series A, CLXXX: 253. 

13 Yule, G. Udny, On the Theory of Correlation for any Number of 
Variables Treated by a new System of Notation, Proc. Roy. Soc, Series 
A, LXXIX: 182. 

i*Kelley, T, L., Educational Guidance, Columbia University, Contribu- 
tions to Education, Teachers College Series, No. 71, 1914, and Tables for 
Facilitating the Calculation of Partial Coefficients of Correlation, etc., 
Univ. of Texas Bulletin, 1916, No. 27. 



68 Tests of Mathematical Ability and Their Prognostic Value 

TABLE XIII 

Weights Given to the Tests Included in the Composite 

FOR Mathematical Ability 

Wadleigh High School 



Algebraic Computation (1 

(2 
Matching Equations and Problems (1 

(2 
Matching iVth Terms and Series.. (1 

(2 
Interpolation (1 

(2 
Missing Steps in Series (1 

(2 
Inference with Symbols (1 

(2 
Geometry (1 

(2 
Superposition (1 

(2 
Symmetry (1 

(2 
Geometrical Definitions (1 

(2 



•^1 

•2 '^ 
Co Q 

7.03 
6.57 
3.13 
3.55 
4.33 
3.62 
8.14 
4.55 
3.03 
3.39 
1.73 
1.74 
5.25 
4.99 
5.69 
6.22 
5.43 
6.55 
5.58 
4.85 



^ s s 

& o S 
^ Co Q 



6.80 



3.34 



3.97 



6.34 



3.21 



1.73 



5.12 



5.95 



5.95 



5.21 






"a 

:5 



The Analysis of Mathematical Ability 69 

Matching Solids and Surfaces (1) 6.19 

5.92 2 4 

(2) 5.65 

Reasoning (1) 3.89 

3.63 1 3 

(2) 3.37 

Arithmetic Problems (1) 1.36 

1.31 2 20 

(2) 1.27 

Multiple equals the number by which the deviations of the test concerned were 
multiplied, in order to give it the desired weight. 

TABLE XIV 
Horace Mann School 

I ^ ^ I ^ '^ •? -i 

Algebraic Computation (1) 6.29 

6.78 2 6 

(2) 7.28 

Matching Equations and Problems (1) 4.95 

6.31 2 6 

(2) 7.67 

Matching i\rth Terms and Series.. (1) 5.28 

5.77 1 4 

(2) 6.26 

Interpolation (1) 37.77 

39.48 2 1 

(2) 41.20 

Missing Steps in Series (1) 2.15 

2.24 1 9 

(2) 2.34 

Inference with Symbols (1) 11.48 

11.55 1 2 

(2) 11.62 

Geometry (1) 5.00 

5.14 2 8 

(2) 5.29 



70 Tests of Mathematical Ability and Their Prognostic Value 

Superposition (1) 5.82 

6.60 1 3 

(2) 7.39 

Symmetry (1) 8.69 

9.18 1 2 

(2) 9.68 

Geometrical Definitions (1) 8.06 

8.04 2 5 

(2) 8.03 

Matching Solids and Surfaces (1) 6.02 

5^ 1 3 

(2) 5.91 

Reasoning (1) 3.78 

3.50 1 6 

(2) 3.22 

Arithmetic Problems (1) 1.37 

1.36 2 30 

(2) 1.35 

Multiple equals the number by which the deviations of the test concerned were 
multiplied, in order to give it the desired weight. 

The coefficients of reliability for the composites for the two 
groups were calculated in the usual manner and are given in 
Tables XV and XVI along with the reliability coefficients for com- 
posites of algebraic, geometrical and verbal ability, which were 
also made on similar lines for use in another connection. 

TABLES XV AND XVI 

Reliability Coefficient for Each Composite and for its 
Two Applications Combined 

Wadleigh Horace 

High School Mann School 

rl r2 r\ r2 

Mathematical ability 86 .92 .93 .96 

Algebraic ability 78 .88 .93 .96 

Geometrical ability 76 .86 .89 .94 

Verbal ability 71 .83 .75 .86 



The Analysis of Mathematical Ability 71 

By means of this standard we can now ascertain the order of 
correlation of each test with mathematical capacity and so deter- 
mine the relative worth of each test as a measure of that ability. 
The results of our calculation are presented in Tables XVII, 
XVIII, XIX, XX, XXI, XXII, which give the values of the 
coefficients both crude and corrected for each group separately 
and for the two groups combined. The crude coefficients are 
more significant for practical diagnosis, but their corrected values 
probably give a more true measure of the amount of correspond- 
ence that exists between each function tested and mathematical 
ability. 



TABLE XVII 

Coefficients of Correlation Between Each Test with the Composite 

FOR Mathematical Ability Arranged in Order 

OF Magnitude (Crude) 

Wadleigh High School 

Mathematical Tests 

r 

Algebraic Computation 55 

Interpolation 55 

Superposition 52 

Missing Steps in Series 49 

Geometry 49 

Matching Equations and Problems 49 

Symmetry 45 

Reasoning 41 

Arithmetic Problems 39 

Matching Solids and Surfaces 38 

Geometrical Definitions 30 

Matching iVth Terms and Series 23 

Inference with Symbols .22 

Verbal Ability Tests 

Trabue Language Scales 39 

Logical Opposites 38 

Mixed Relations 23 

Thorndike Reading Scales 22 



72 Tests of Mathematical Ability and Their Prognostic Value 

TABLE XVIII 

Coefficients of Correlation Between Each Test with the Composite 

FOR Mathematical Ability Arranged in Order 

OF Magnitude (Crude) 

Horace Mann School 

Mathematical Tests r 

Algebraic Computation 76 

Interpolation 72 

Missing Steps in Series 70 

Geometry 69 

Arithmetic Problems 61 

Matching Equations and Problems 61 

Geometrical Definitions 60 

Superposition 60 

Inference with Symbols 58 

Reasoning 53 

Matching Solids and Surfaces 48 

Symmetry 48 

Matching Nth. Terms and Series 41 

Verbal Ability Tests 

Reading — Understanding of Sentences 47 

Mixed Relations 47 

Logical Opposites 42 

Trabue Language Scales ZZ 

TABLE XIX 
Wadleigh High School and Horace Mann School (Combined) 

Mathematical Tests r P.E. 

Algebraic Computation 69 .05 

Interpolation 66 .04 

Missing Steps in Series 63 .06 

Geometry 6Z .05 

Superposition 57 .02 

Matching Equations and Problems 57 .03 

Arithmetic Problems 54 .06 

Geometrical Definitions 50 .07 

Reasoning 49 .03 

Symmetry 47 .07 

Inference with Symbols 46 .09 

Matching Solids and Surfaces 45 .03 

Matching Nth. Terms and Series .35 .04 

Verbal Ability Tests 

Logical Opposites 41 .01 

Thorndike Reading Scales 39 .06 

Mixed Relations .39 .06 

Trabue Language Scales 35 .01 



The Analysis of Mathematical Ability 73 

TABLE XX 

Coefficients of Correlation Between Each Test with the Composite 

FOR Mathematical Ability Arranged in Order 

of Magnitude (Corrected) 

Wadleigh High School 

Mathematical Tests 

r 

Interpolation 70 

Matching Solids and Surfaces 68 

Reasoning 68 

Algebraic Computation 67 

Matching Equations and Problems 66 

Geometry 64 

Arithmetic Problems 62 

Superposition 62 

Missing Steps in Series 61 

Geometrical Definitions 53 

Symmetry • 50 

Inference with Symbols 47 

Matching Nth. Terms and Series 31 

Verbal Ability Tests 

Trabue Language Scales 62 

Logical Opposites 51 

Thorndike Reading Scales 32 

Mixed Relations 23 



TABLE XXI 

Horace Mann School 

Mathematical Tests 

r 

Algebraic Computation 87 

Missing Steps in Series 87 

Geometry 83 

Matching Equations and Problems 81 

Arithmetic Problems 80 

Interpolation 77 

Superposition 68 

Geometrical Definitions 68 

Inference with Symbols 66 

Reasoning 64 

Matching Solids and Surfaces 59 

Symmetry 51 

Matching Nth Terms and Series 46 



74 Tests of Mathematical Ability and Their Prognostic Value 

Table XXI— Continued 
Verbal Ability Tests 

Reading — Understanding of Sentences 6Z 

Logical Opposites 57 

Mixed Relations 55 

Trabue Language Scales 54 



TABLE XXII 

Coefficients of Correlation Between Each Test with the Composite 

FOR Mathematical Ability Arranged in Order 

OF Magnitude (Corrected) 

Wadleigh High School and Horace Mann School (Combined) 

Mathematical Tests 

r P.E. 

Algebraic Computation 81 .05 

Missing Steps in Series 78 .06 

Geometry 76 .05 

Matching Equations and Problems 76 .04 

Interpolation 75 .02 

Arithmetic Problems 74 .04 

Superposition 66 .01 

Reasoning 65 .01 

Geometrical Definitions 63 .04 

Matching Solids and Surfaces 62 .07 

Inference with Symbols 60 .05 

Symmetry .51 .03 

Matching iVth Terms and Series 41 .04 

Verbal Ability Tests 

Trabue Language Scales 57 .02 

Logical Opposites 55 .01 

Thorndike Reading Scales 53 .08 

Mixed Relations 44 .08 



The uniformity in the results obtained will be readily recog- 
nized. When the observed quantities of correlation for the va- 
rious tests are compared with their probable error, such differences 
in rank as are found between the two applications appear negligi- 



The Analysis of Mathematical Ability 75 

ble. It will be profitable at this point to compare with the above 
results the relative positions of the tests, when arranged in the 
order of the magnitude of their correlation with mathematical 
ability, using our third standard. These have been computed 
from the amalgamated results of the two groups for both crude 
and corrected coefficients. They are included in Table XXII I. 



TABLE XXIII 

Average Correlation of Each Test with Every Test Arranged in 

Order of Magnitude: Wadleigh High School and 

Horace Mann School Combined 

Crude Coefficients'. 

r 

Algebraic Computation Z6Z 

Missing Steps in Series 349 

Geometry 346 

Interpolation 345 

Superposition 323 

Matching Equations and Problems 300 

Geometrical Definitions 285 

Reasoning 273 

Arithmetic Problems 268 

Symmetry 258 

Inference with Symbols 250 

Matching Solids and Surfaces 249 

Matching Nth. Terms and Series 195 

Corrected Coefficients: 

Missing Steps in Series 478 

Algebraic Computation 466 

Geometry 457 

Superposition .435 

Interpolation 432 

Arithmetic Problems 429 

Matching Equations and Problems 426 

Reasoning 392 

Inference with Symbols 364 

Geometrical Definitions 360 

Matching Solids and Surfaces 345 

Symmetry 298 

Matching Nth. Terms and Series 228 



76 Tests of Mathematical Ability and Their Prognostic Value 

The order of the tests corresponds closely to that obtained 
from the application of the second standard, but the differences 
in the coefficients are extremely small and suggest a chance dis- 
tribution. On the other hand, by means of the first standard used 
we can decide between the tests as more or less indicative of 
mathematical intelligence. Thus of the thirteen tests of alge- 
braic and geometrical abilities, seven yield coefficients (Table 
XXII) above .65, while six give values below .65. Obviously 
the former tests probe certain characteristics more fundamental 
in mathematical efficiency than the latter. It is equally clear, 
however, that no single test is a sufficient index to mathematical 
mastery. Even when the coefficients have been corrected for ac- 
cidental errors of measurement, no test correlates perfectly with 
the composite. When it is remembered that the latter includes 
these tests now correlated with it, this fact will be all the more 
striking.^^ Among the highest observed raw coefficients for the 
two groups combined is that between the composite for mathe- 
matical ability and Algebraic Computation.^^ Its value is only 
.69. Moreover, the Symmetry test ^^ (to take only one instance), 
with which apparently it has little observable relationship (.17) 
correlates with the composite ^^ to the extent of .47. The corres- 
ponding corrected coefficients ^^, ^°, ^^ are .81, .19, and .51. It 
seems that mathematical ability is a complex resultant of many 
loosely knit capacities, all working together. 

Further light is thrown upon its nature by a consideration of 
the coefficients obtained in the case of the tests of verbal ability. 
These tests, it has to be remembered, unlike the mathematical 
tests, were not included in the composite. Their value is notably 
high. To make one or two comparisons from the corrected and 
combined coefficients of Wadleigh and Horace Mann, greater 
kinship apparently exists between mathematical ability and the 

15 The relationship of each test and the composite independent of its 
own contribution to the latter could be determined by the use of Partial 
Coefficients of Correlation. Each test, however, measures the efficiency 
of an activity, which has a prima facie claim to be called mathematical 
and therefore its influence in the composite ought to be expressed in the 
coefficient denoting the correspondence between mathematical ability and 
the function measured by it. The positive correspondence due to this 
is not properly described as spurious correlation, 

16 See Table XIX. i^ See Table XXII. 

17 See Table IX. 20 See Table XII. 

18 See Table XIX. 21 See Table XXII. 



The Analysis of Mathematical Ability yy 

functions measured by the Trabue Language Scale, the Thorn- 
dike Reading Tests and the test of Logical Opposites than between 
the former and the functions measured by the Matching Nth. 
Terms and Series or the Symmetry tests. (See Tables XIX and 
XXIL) 

Other instances of the influence of ability with words upon 
ability in algebra and geometry can be had from an analysis of 
the tests which involve similar mental acts, bVt differ as regards 
content. Such a group of tests are the Interpolation test, the 
Missing Steps in Series test and the Trabue Language Scales. 
Each of these demands an anlysis of the given facts and their 
supplementation or completion in such a way as to produce a 
coherent and inclusive whole. The differences lie merely in the 
material. In the first and second of these the data are numbers ; 
in the third, the data are words. The correspondences found be- 
tween these functions show how important a role language plays 
in these performances. Between the Interpolation test and the 
Missing Steps in Series test the coefficient of correlation amounted 
to .57 (crude) ^^ while between the Interpolation test and the 
Trabue Language Scales it was only .14 (crude). ^^ Similarly be- 
tween the Missing Steps in Series test and the Trabue Language 
Scales the correlation observed was .21 ( crude ).^^ The corres- 
ponding coefficients, when corrected, became .71, .18, and .42 re- 
spectively.^^ 

Again Matching Problems and Equations, Matching Nth. Terms 
and Series and Matching Solids and Surfaces demand similar 
mental processes of analysis and identification, while diifering 
considerably in content. In the first pair (one involving words 
and numbers, and the other involving numbers only) the corres- 
pondence amounted to .26,^* while between Matching Problems 
and Matching Solids and Surfaces and between Matching Mh 
Terms and Matching Solids and Surfaces,^^ there was .09 and .03 
respectively. The corresponding corrected coefficients ^® were .35, 
.12, and .09. 

More striking still is the evidence of the complexity of mathe- 
matical capacity, which follows from grouping the tests of kindred 
nature. When a composite of algebraic ability is made in the 

22 See Table IX. 

23 See Table XII. 2^ See Table IX. 25 /^,f^. 26 See Table XII. 



y8 Tests of Mathematical Ability and Their Prognostic Value 

same way as described earlier in this study, weights being given 
in identical proportions to the various tests and when corres- 
ponding composites are made for geometrical ability and verbal 
or language ability, results of considerable interest are obtained. 

Before presenting these, however, the method in which the 
composite for verbal ability was constructed will be shown. In 
Tables XXIV and XXV the weights given to the tests are indi- 
cated. The reliability coefficients for these new composites were, 
in the case of the Wadleigh High School,^^ .78, .76 and .71 for 
algebraic, geometrical and verbal ability, and similarly for Horace 
Mann School,^^ they were .93, .89 and .75 respectively. 

Tables XXVI, XXVII and XXVIII present the various coef- 
ficients of correlation between these composites both crude and 
corrected, and separately and combined. 

TABLE XXIV 

Weights Given to Each Test in the Composite for Verbal Ability 
Wadleigh High School 



Mixed Relations (1) 5.09 

4.74 1 3 

(2) 4.39 

Logical Opposites (1) 7.78 

6.97 1 2 

(2) 6.17 

Trabue Language Scales (1) 2.62 

2.84 2 10 

(2) 3.07 

Thorndike Reading Scales (1) 2.91 

3.56 2 8 

(2) 4.22 

Multiple equals the number by which the deviations of the test concerned were 
multiplied, in order to give it the desired weight. 

27 See Table XV. 

28 See Table XVL 



The Analysis of Mathematical Ability 79 

TABLE XXV 

Weights Given to Each Test in the Composite for Verbal Ability 
Horace Mann School 



go <« S <^ 



cs '-2 ^ o _ 



•0 *» -:r 



Mixed Relations (1) 9.62 

(2) 871 

Logical Opposites (1) 19.39 

(2) 23.02 

Trabue Language Scales (1) 3.20 

(2) 4.09 

Thorndike Reading Scales (1) 2.77 

(2) 8.22 



9.16 



21.20 



3.64 2 12 



5.49 



Multiple equals the number by which the deviations of the test concerned were 
multiplied, in order to give it the desired weight. 

TABLE XXVI 

Coefficients of Correlation Between the Composites for Mathematical 

Ability, Algebraic Ability, Geometrical Ability 

AND Verbal Ability 

Wadleigh High School 

Crude 

r 

Mathematical ability and Verbal ability 44 

Algebraic ability and Geometrical ability 38 

Algebraic ability and Verbal ability 42 

Geometrical ability and Verbal ability 41 

Corrected 

r 

Mathematical ability and Verbal ability 57 

Algebraic ability and Geometrical ability 49 

Algebraic ability and Verbal ability 56 

Geometrical ability and Verbal ability 56 



8o Tests of Mathematical Ability and Their Prognostic Value 

TABLE XXVII 

Coefficients of Correlation Between the Composites for Mathematical 

Ability, Algebraic Ability, Geometrical Ability 

AND Verbal Ability 

Horace Mann School 

Crude 

r 

Mathematical ability and Verbal ability 58 

Algebraic ability and Geometrical ability 52 

Algebraic ability and Verbal ability 1 48 

Geometrical ability and Verbal ability 49 

Corrected 

r 

Mathematical ability and Verbal ability 69 

Algebraic ability and Geometrical ability 57 

Algebraic ability and Verbal ability 57 

Geometrical ability and Verbal ability .61 

TABLE XXVIII 

Coefficients of Correlation Between the Composites for Mathematical 

Ability, Algebraic Ability, Geometrical Ability 

and Verbal Ability 

Wadleigh High School and Horace Mann School (Combined) 

Crude 

r P.E. 

Mathematical ability and Verbal ability 54 .01 

Algebraic ability and Verbal ability 46 .01 

Geometrical ability and Verbal ability 47 .07 

Algebraic ability and Geometrical ability 47 .03 

Corrected 

r P.E. 

Mathematical ability and Verbal ability 65 .09 

Algebraic ability and Verbal ability 57 .05 

Geometrical ability and Verbal ability 59 .01 

Geometrical ability and Algebraic ability 54 .02 



The Analysis of Mathematical Ability 8i 

The closeness of relationship between mathematical ability and 
ability with words as represented in these four tests, Mixed Re- 
lations, Logical Opposites, Trabue Language Scales, and the 
Thorndike Reading Tests is important. Several writers have 
pointed out, notably Suzzallo,"^ that in primary arithmetic the 
problem of teaching children to reason is largely a matter of teach- 
ing children language and how to use it. ''Reasoning in school 
problems has far more to do with the language involved in a 
problem than with the numbers or combinations of numbers." 
It would appear that this still remains true in later mathematical 
work. The ability to understand sentences, to conceive clearly 
the meaning of a given problem, is as important an element in its 
solution as any connection it has with algebraic symbols or their 
manipulation. The relative significance of ability with language 
is made apparent by the degree of correlation that these coefficients 
reveal. 

It would thus appear that the tests in verbal ability enable us 
to prophesy efficiency in algebra with as great a chance of suc- 
cess as the tests in geometry would and contrariwise that they are 
as reliable indices of the characteristics which make a successful 
student of geometry, as the algebra tests are. 

These results raise the important question of what the true 
relationship between algebraic ability and geometrical ability is, 
when these are freed from this common factor, verbal ability. 
How far is the correlation between algebraic and geometrical 
ability due to the correlation that exists between each of these 
and the abilities measured by the tests of what we have called 
verbal ability and to what extent is it independent of the latter? 

To find an answer to this, recourse was had to the method of 
Partial Coefficients of Correlation, by which the relationship be- 
tween two functions for a constant value of a third can be deter- 
mined. 

29 Suzzallo, H., Reasoning in Primary Arithmetic, California Education, 
June, 1906: 189. 



82 Tests of Mathematical Ability and Their Prognostic Value 
The formula used was the following : ^^ 



'12 '13 '23 

'19«9 — — ——————————— 



V(l-r^i.3) (1-^.-3) 

in which r^z-s indicates the correlation between traits 1 and 2, 
for a constant value of trait 3. The reasoning underlying the 
Partial Correlation formula for three variables can be simply 
illustrated. Suppose that of the sixty-one Horace Mann students 
examined, ten are of approximately equal capacity in the verbal 
ability tests. The achievements in algebra and geometry of this 
group, in which verbal ability is constant, are then correlated. 
The resulting coefficient gives the partial correlation of algebraic 
and geometrical abilities for a constant value of verbal ability; 
that is, it expresses the relation of ability in algebra to ability 
in geometry independent of language ability; or, in other words, 
it represents the extent to which these abilities are related apart 
from their common connection with the ability to deal with words. 

The application of this method gave significant results. Let 
us consider the crude coefficients. For the one group tested the 
new relationship between algebraic ability and geometrical ability 
finds expression in the coefficient .25 ± .07. The corresponding 
result for the other group is .37 ± .05. Averaging these coef- 
ficients and attaching double weight to the Horace Mann figures 
in accordance with our usual procedure, the value of the coef- 
ficient of correspondence between the two abilities obtained for 
the total number of persons tested is .33 it .03. 

Undoubtedly these partial coefficients do not represent the pure 
relationship between algebraic ability and geometrical ability in 
isolation from the influence of their mutual relations with all 
other traits. With the elimination of the latter there would be 
still further reduction in the coefficient. It should, be remembered 
also that irrelevant factors such as age certainly affect the de- 
gree of correspondence observed. The correlations of the various 
composites with age and their interrelations after the effect of age 

30 Yule, G. Udny, An Introduction to the Theory of Statistics, London, 
1916. Chapter 12. 



The Analysis of Mathematical Ability 83 

has been eliminated are interesting in this connection. These are 
summarized in Tables XXIX, XXX, and XXXI. 



TABLE XXIX 

Coefficients of Correlation Between the Composites for Mathematical 
Ability, Algebraic Ability, Geometrical Ability, and Verbal 
Ability and Age, for Wadleigh High School, Horace Mann 
School and Both Combined 

Crude Coefficients 

W.H.S. & 
W.H.S. H.M.S. H.M.S. 
Mathematical ability and Age.. — .30 — .48 — .42 

Algebraic ability and Age — .36 — .52 — .47 

Geometrical ability and Age. . [ — .04] — .Z3 — .21 

Verbal ability and Age [—-07] —.35 —.26 

Diagnostic composite and Age — .27 — .52 — .44 

Coefficients less than 2 P.E. are put in square brackets. 



TABLE XXX 

Corrected Coefficients 

W.H.S. & 

W.H.S. H.M.S. H.M.S. 

Mathematical ability and Age.. — .32 — .50 — .44 

Algebraic ability and Age — .41 — .53 — .49 

Geometrical ability and Age.. — .35 

Verbal ability and Age —.03 —.40 —.27 

Diagnostic composite and Age — .29 — .55 — .46 



TABLE XXXI 

Coefficients of Correlation Between the Composites for Mathematical 
Ability, Algebraic Ability, Geometrical Ability and Verbal 
Ability, the Effect of Age Being Eliminated (Crude) 

Wadleigh High School, Horace Mann School and Both Combined 

W.HS. & 
W.H.S. H.M.S. H.M.S. 

Mathematical ability and Verbal ability 44 .50 .48 

Algebraic ability and Geometrical ability. . . .39 .43 .42 

Algebraic ability and Verbal ability 43 .Z7 .39 

Geometrical ability and Verbal ability 41 .42 .42 



84 Tests of Mathematical Ability and Their Prognostic Value 

By the application of the same method of Partial Coefficients 
of Correlation it is possible to ascertain the relation between alge- 
braic ability and geometrical ability independent of ability with 
words and unaffected by age. In the case of the Wadleigh High 
School students the coefficient determined from the crude values 
was .26, for the Horace Mann students it was .33 and for these 
combined in the usual manner it was .30. Our results thus con- 
firm those obtained by Brown '^ that algebra and geometry de- 
mand activities of different kinds, although algebraic and geo- 
m.etrical abilities are positively related as is usual in the case of 
desirable traits. They lend no support to the view that there is a 
"special capacity or faculty underlying mathematical ability, dis- 
tinct from and with no essentially close connections with other 
forms of mental capacity." 

The results so far obtained lend further support to the view 
that mathematical intelligence is complex in character, embracing 
a variety of mental processes, which are somewhat loosely re- 
lated, but equally indispensable for successful accomplishment in 
the subject. A more penetrating analysis of its general nature 
has still to be made. Can our results afford any explanation of 
such correspondences as have been found ? Can they suggest the 
characteristics in the tests which make for high correlation with 
mathematical ability? Is there any common psychological factor 
in those tests which correlate closely with mathematical ability? 
Are they saturated, as it were, with some quality which pervades 
them in different amounts? Does some feature common to the 
Algebraic Computation, Interpolation, Missing Steps in Series 
and the Geometry Tests explain why these tests correlate more 
closely with the composite than do the remaining tests ? 

We need not consider at this time such effects as that of general 
technique of administration, which undoubtedly exercises an im- 
portant influence upon the correlations observed. For example, 
the mere duration of the time of testing (the lengthiness or short- 
ness of the test) has a marked effect. Where the tests are sup- 
posedly given equal weight in the composite, in fact each is given 
a weight in rough proportion to the time of testing. When we 
review several of the explanations that might be given in answer 

31 See Brown, William, An Objective Study of Mathematical Intel- 
ligence, Biometrika, VII : 361. 



The Analysis of Mathematical Ability 85 

to the questions propounded, such as the amount of novel or 
familiar experience the test entails, or the degree of complexity 
or simplicity it involves, or the demands it makes upon the ability 
to abstract and analyze and think selectively, we are led to the 
conclusion that while all of these, together with many other fac- 
tors, may be determinants of the correlations found between 
mathematical ability and the tests, yet the highest common psy- 
chological factor, which explains the character of the correspon- 
dences revealed, is this ability to react to partial elements in a 
situation rather than to gross totals.^^ While mathematical intel- 
ligence can neither be satisfactorily diagnosed, nor explained by 
reference to a single test or a single mental process, yet the 
experimental evidence we have obtained suggests that a marked 
degree of the power to analyze a complex and abstract situation 
and to seize upon its implications is the most indispensable element 
in mathematical proficiency. The view advanced receives strik- 
ing support in the high correlation between the composite for • 
mathematical ability and the Interpolation test. The latter de- 
mands the ability to analyze abstract elements, to generalize from 
these and further to make application of the principle discovered. 
It is highly symptomatic of mathematical ability, and the factor 
common to it and to the other tests correlating closely with the 
composite is apparently this marked facility in the analysis of 
partial elements in a complex situation presented. The corres- 
pondences we have found suggest that mathematical intelligence 
embraces a wealth of functions, whose psychical nature is hard to 
detect and describe in detail, but if there is any community be- 
tween these, it would seem to be of the kind described. Be- 
sides this general common factor, which is distributed in different 
amounts in all the tests that can be called mathematical, there are 
specific factors of varying importance. Thus in the Geometry 
test the common factor is present in considerable amount, but 
the specific differences between this test and the Interpolation 
test are obviously large. For a complete description of mathe- 
matical ability the latter are essential; but while these factors 
are necessary to success, the former still remains the most char- 
acteristic quality in mathematical mastery and tests possessing it 

32 Thorndike, E. L., Educational Psychology, II: Chap. 4. 



86 Tests of Mathematical Ability and Their Prognostic Value 

in marked degree will serve best as an index to the presence of 
those qualities that determine success with the subject. 

We have still, however, to evaluate both tests and composite by 
means of an independent estimate of the relative mathematical 
efficiency of the individuals tested. For this purpose the school 
marks in algebra and geometry up to date have been utilized. In 
the case of the Wadleigh High School group, two distinct sets of 
measurements were available, but this was lacking in the case of 
the Horace Mann group. Both the standard "Product-Moments" 
method and the Method of Ranks were applied to determine the 
extent to which the composite for mathematical ability was 
diagnostic of mathematical ability as measured by school records. 
The results can be most clearly presented in tabular form. Those 
obtained by the standard method are given first. 



TABLE XXXI 
Pearson Product-Moments Method 

Group Reliability Raw Average Crude Corrected 

Coefficients Coefficients Coefficients Coefficients 

Wadleigh 

High School W^ W2=.88 W-j Sch2=.59 W Sch=.54 W Sch=.70 

Schj Sch2=.87 W^ Sch^=.50 

Horace Mann 

School W., W2=.85 W-, Sch.>=.75 W Sch=.75 W Sch=.88 

J Sch2=.8J 
(assumed) 



Here W^ W2 stand for the two applications of the tests in- 
cluded, in the composite for mathematical ability and Sch^ Sch^ 
stand for the two independent series of school marks. In the case 
of the Horace Mann pupils, the sole available score (an average of 
four grades by the same teacher) was correlated with both ap- 
plications of the composite of mathematical ability. In order 
to correct for attenuation, its reliability was assumed to be .85. 
It will certainly not differ greatly from .85 when calculated from 
the second series of records from Horace Mann, which will later 
be available. 



The Analysis of Mathematical Ability 87 

The Method of Ranks gave the results tabulated below: 



Group 


Reliability 
Coefficients 


Raw 

Coefficients 


Average Raw 

Coefficients 


Corrected 
Coefficients 


Wadleigh 
High School 


W^ W2=.89 
Sch^ Sch2=.80 


Wj Sch4j=.70 
W2 Sch^=.53 


W Sch=.61 


W Sch=.72 


Horace Mann 
School 


W-. W2=.95 
Schj Sch2=.85 
(assumed) 


W,j Sch2=.79 
W2 Schj=.79 


W Sch=.79 


W Sch=.88 



While the crude coefficients are of more significance than their 
theoretical corrections for practical diagnosis, the corrected coef- 
ficients give the most probable true amount of correspondence be- 
tween the functions, which are represented by the two sets of 
measures. Considering the facts from Wadleigh High School, 
in which we have records for two years' work in the subject, we 
note that the composite foretells how well a pupil will do in his 
second year about three-fourths as accurately as does his entire 
record for the first year. 

Using the corrected coefficients, we see that the degree of cor- 
respondence between the functions represented by the school 
marks and those represented by the composite is for the Wad- 
leigh results, .70 and in the case of the Horace Mann results, .88. 
Combining these and giving the Horace Mann results double 
weight on account of their greater reliability, we find that the 
coefficient of correlation between school marks and the composite 
for mathematical ability is .82, a closeness of correspondence 
rarely found either in mental or physical measurements. 

It is evident that our composite does measure the abilities 
fundamental to success in High School Mathematics, as measured 
by school records. The above coefficients indicate that in the 
composite we have a measure of the mental functions that are 
at work in the activities of the mathematics class. Accordingly 
our analysis of the characteristics that explain the correspondences 
observed between the traits measured by the tests is essentially 
an analysis of the qualities that determine successful accomplish- 
ment in the class room. Further, it has to be taken into consider- 
ation in judging of the kinship between the abilities covered by 



88 Tests of Mathematical Ability and Their Prognostic Value 

the tests and the abilities functioning in school records that the 
two groups of subjects examined were unusually homogeneous in 
character as a result of an ^uncommonly careful and successful 
system of grading. Thus classification of pupils into groups of 
approximately equal ability was much more nearly attained in 
the case of these students than is at all usual. This tended to re- 
duce the amount of correlation between school records and the 
tests and undoubtedly where a group of average variation is ex- 
amined the aid the tests can lend towards classification will be 
yet greater. 

Even in the case of these well graded schools, however, the 
difference in achievement of the various individuals tested was 
considerable. This indicates that the tests would prove of value 
in allocating pupils to a suitable group. In particular where our 
purpose is to select from a large number a class that can make 
rapid headway in this subject or at the other extreme a group 
whose likelihood of success is slight, the tests can afford invalu- 
able aid in relegating each to his proper place. This would not 
only be of great benefit to the particular individuals concerned, 
but would put a difficult administrative task on a scientific foot- 
ing. Thus the tests provide a means of measuring mathematical 
intelligence. By their aid we can determine in advance within 
known limits the relative standing of any individual in the subject. 
We can prophesy with a known degree of certainty from suc- 
cess with the tests a successful record in school work and contrari- 
wise we can predict failure in the mathematics course from failure 
with them. 

A further consideration which needs emphasis is that some of 
the tests owe their value to the very fact that they measure abilities 
which the school examinations fail to measure, but which are in- 
dispensable for success in future mathematical work. We would 
not expect perfect correspondence, therefore, between the com- 
posite for mathematical ability and the school records. Among 
such tests we would include the Matching Solids and Surfaces 
test, the Missing Steps in Series test, the Interpolation test and 
perhaps the Superposition test. These involve somewhat special- 
ized traits. Such abilities as skill with spatial data, ability to 
think with space and to manipulate novel symbols, are important 
factors in future success. These are but poorly measured by 



The Analysis of Mathematical Ability 89 

school examination marks and yet they are quaHties crucial in the 
mastery of the subject. Inasmuch as the tests supply this lack, 
they offer valuable supplementation to the ordinary school- 
methods of measurement. 



CHAPTER IV 
THE PROGNOSIS OF MATHEMATICAL ABILITY 

We can thus bring the foregoing experimental results and theo- 
retical conclusions to bear upon this last and most important of 
the problems with which this study is concerned: the practical 
prognosis of mathematical ability. The task in essence is to 
choose from the total list of tests a group, economical of time, 
easy of application, and possessing maximum diagnostic power. 
In making this selection our previous analysis of the factors that 
contribute towards success will lend invaluable aid. 

Certain general principles serve to guide our choice in addition 
to the above practical considerations. Since the tests should 
cover as many phases of mathematical capacity as possible, those 
tests preeminently should, be chosen which are both closely cor- 
related with the composite for mathematical ability, and loosely 
correlated with each other. Guided by such experimental evi- 
dence of their nature as this study provides, together with existing 
knowledge of the characteristic qualities of those tests which have 
already been extensively used, we can provisionally select a set 
of tests whose value can later be established by the degree to 
which the results of their application coincide with a reliable in- 
dependent estimate of the mathematical ability of the individuals 
examined. 

Six tests were selected from the available seventeen to make 
this new Diagnostic Composite. These were the following: 
Algebraic Computation, Interpolation, Geometry Test, Superposi- 
tion,^ Mixed Relations, and the Trabue Language Scales. 

Tables XXXII and XXXIII show the method by which the 
Diagnostic Composite was constructed. 

1 Modified Thurstone's Spacial Relations Test. 

90 



The Analysis of Mathematical Ability 91 

TABLE XXXII 

Weights Given to the Tests Included in the Diagnostic Composite 
FOR Mathematical Ability 

Wadleigh High School 

^q ^<>5q qs :^ 

Algebraic Computation (1) 7.03 

6.80 2 4 

(2) 6.57 

Interpolation (1) 8.14 

6.34 2 4 

(2) 4.55 

Geometry (1) 5.25 

5.12 2 5 

(2) 4.99 

Superposition (1) 5.69 

5.95 1 2 

(2) 6.22 

Mixed Relations (1) 5.09 

4.74 1 3 

(2) 4.39 

Trabue Language Scales (1) 2.62 

2.84 1 5 

(2) 3.07 



92 Tests of Mathematical Ability and Their Prognostic Value 

TABLE XXXIII 

Weights Given to the Tests Included in the Diagnostic Composite 
FOR Mathematical Ability 

Horace Mann School 



■il t-§| 1l> i 

^q ^C^o q^ ^ 

Algebraic Computation (1) 6.29 

6.78 2 6 

(2) 7.28 

Interpolation (1) Z7.77 

39.48 2 1 

(2) 41.20 

Geometry (1) 5.00 

5.14 2 8 

(2) 5.29 

Superposition (1) 5.82 

6.60 1 3 

(2) 7.39 

Mixed Relations (1) 9.62 

9.16 1 2 

(2) 8.71 

Trabue Language Scales (1) 3.20 

3.64 1 6 

(2) 4.09 



The Analysis of Mathematical Ability 



93 



It will be noted that these tests have been taken from each of 
the three main groups of abilities investigated, since our results 
have shown that each of these plays an important role in mathe- 
matical work. Further within these groups, tests correlating 
loosely with each other and closely with the composite for mathe- 
matical ability have been selected, wherever the subsidiary re- 
quirements of economy of time and ease of administration were 
fulfilled. The above sextet of tests can be applied in an hour 
and a half.^ To prove the wisdom of our choice the resulting 
composite measures were correlated with the school marks ob- 
tained by each individual. The coefficients derived are summa- 
rized in the tables that follow : 



Pearson Product- Moments Method 



Group 



Reliability 
Coefficients 



Crude 
Coefficients 



Average Raw 
Coefficients 



Corrected 
Coefficients 



Wadleigh 
High School 


Ai A2=-87 
Sch^ Sch2=.70 


Ai Sch2=.56 
A 2 Sch-^=.68 


A Sch = .62 


A Sch=.79 


Horace Mann 
School 


Ai A2=-92 

Sch^ Sch2=.85 
(assumed) 


A^ Sch=.94 
A2 Sch=.71 


A Sch = .82 


A Sch=.92 



Ai and Ag represent the two applications of the tests included 
in the Diagnostic Composite for mathematical ability. Sch^ 
and Scha represent two independent series of school marks. As 
before, only one mark was available in the case of the Horace 
Mann girls. 

Method of Ranks 



Group 


Reliability 
Coefficients 


Crude 
Coefficients 


Average Crude 
Coefficients 


Corrected 
Coefficients 


Wadleigh 
High School 


Ai A2— '^^ 
Sch^ Sch2=.80 


Ai Sch2=.61 
A2 Scl-^=.50 


A Sch=.55 


A Sch=.65 


Horace Mann 
School 


Ai A2=-91 

Scl^ Sch2=.85 
(assumed) 


Ai Sch2=.78 
A2 Schj=. 77 


A Sch=. 77 


A Sch=.88 



2 A simple plan for the application and evaluation of these tests is 
given in the Appendix. 



94 Tests of Mathematical Ability and Their Prognostic Value 

These figures offer substantial evidence of the significance of 
the six tests composing the Diagnostic Composite as measures of 
promise of mathematical performance in high school. The re- 
liability of the composite is high and its prognostic power is such 
that in the case, for example, of the Wadleigh High School pupils 
one-half of the school record can be predicted with almost as great 
accuracy from the tests as from the other half. Thus an hour 
and a half spent on the tests may be expected to give a correla- 
tion of from .60 to .80 with future mathematical achievement. 
By means of these half dozen tests we are able to grade a group 
of pupils in an order of ability in mathematics and to classify 
them. 

Further we are enabled to diagnose the lines of strength and of 
weakness in an individual's equipment for the subject, to dis- 
cover, for instance, whether feeble intuitive grasp of spatial re- 
lations is the reason for failure with solid geometry, whether 
a poor command of language is the cause of lack of success with 
algebra problems. The tests are far from doing so with per- 
fect precision. Investigation with many more tests upon a larger 
number of subjects would undoubtedly yield better methods of 
measurement. It is not only desirable to extend the tests and to 
supplement them, but to try others. Nevertheless, even in their 
present form they will prove useful, for by their aid we can pre- 
dict with a known degree of accuracy the capacity of the pupil to 
undertake the high school course in mathematics. Not only be- 
cause they measure abilities untested by ordinary examinations 
and important for success in the study of the subject, not only be- 
cause they ascertain the ability of the pupil in greater detail, locat- 
ing weaknesses or talent, but far more because they are exact 
measures, objective measures, which another can repeat and con- 
firm or refute, they show themselves superior to the ordinary class 
examination and have a claim to consideration. They certainly 
will not have the same probable error and low reliability coef- 
ficients that characterize school marks. When we consider the 
results of Starch and Elliott's ^ investigation into the reliability 
of grading work in mathematics, the wide variation of grades 
given to the same paper by different teachers must arouse distrust 

3 Starch, D. and Elliott, E. C, Reliability of Grading Work in Mathe- 
matics, School Review, XXI: 254. 



The Analysis of Mathematical Ability 95 

of conclusions founded upon such faulty data. Conclusions can- 
not be more trustworthy than the figures upon which they are 
based. The ordinary examination does not attempt to satisfy 
the conditions that the tests partially realize. The Diagnostic 
Composite can in an hour and a half provide a reasonably ob- 
jective measure of the mathematical ability of the individual. 



CHAPTER V 
SUMMARIZED CONCLUSIONS 

I. The crude coefficients of correlation between mathematical 
abilities for both groups combined range from .01 to .59. The 
corrected coefficients of correlation between mathematical abilities 
for both groups combined range from .06 to .82. 

II. When mathematical ability is represented by a composite 
of all the mathematical tests, the highest correlation between that 
composite and any test is for the crude values .69 ± .05, and for 
the corrected values .81 ± .05. 

III. The six best measures of mathematical ability, together 
with their correlations with the composite, are : 

Crude r P.E. Corrected r P.E. 

Algebraic Computation . . .69 .05 Algebraic Computation . . .81 .05 

Interpolation 66 .04 Missing Steps in Series.. .78 .06 

Missing Steps in Series.. .63 .06 Geometry .76 .04 

Geometry 63 .05 Matching Equations and 

Superposition 57 .02 Problems 76 .04 

Matching Equations and Interpolation 75 .02 

Problems 57 .03 Arithmetic Problems 74 .04 

IV. No single test is a sufficient index to mathematical ability. 

V. The functions represented by the three groups of tests 
for algebraic, geometrical, and verbal abilities are all equally 
essential in mathematical ability. The correlations between the 
composite for mathematical ability and each of these three groups 
of tests are practically the same. 

VI. The correspondences found between the mathematical 
abilities tested may be traced to the common characteristic of 
capacity to react to partial elements in a situation. Mathematical 
ability is the complex resultant of a number of loosely connected 
capacities. 

VII. Mathematical ability can be satisfactorily diagnosed by 
six tests requiring an hour and a half in time. 

96 



APPENDIX 



98 Tests of Mathematical Ability and Their Prognostic Value 



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(X) S3TJ3g puB SUJJ3X ^^5<l Suiqo;Bjij[ 



(^) sraayqojfj puB suoi^Bnbg; Suiqo^Bj^j 



(X) sui9tqoi(j puB suoT^Bnbg; SuiqD;Bj\[ 



^^3) uoi;B;ndmo3 oiBjqaSiy 



(X) uoi;B;ndui03 oiBjq9S|V 



f5 0\rorcr1 
OQ C^ >-l CVl rl 


t^CMCVI0\O 


CSirororH^ 


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r^THTi ,-1 


rH CM CN) r<^ CM 


25:5^^2 


t^t^rt OOfO 
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io<r}ioO'-i 

CNXNrH 


10 00 eg t^ CO 

rO <N1 CVl r-l (M 


^^2^2 


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t^C^rHfOON 


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rO CM i-H rH r-H 


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inc<i\oiooo 

<M CSI (N r-H-H 


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CM <M r-l r-l .-1 


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00000 


vot-^ooo\o 
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1-1 C^3 ro Tf u^ 


vot^oocyso 



Appendix 



103 



OOOCht^Tf 



fOT-Ht^rHoo cooMoOo ^^o^oolo\o oorxi-ivo>-i 00 "-i c'j ti- t^ oo\^'OTf 00 vo cm t-n ti ^^<mcc\oo\ 



<M\0 



vovo 

T-Iod 



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OOOt^r-iOO 00»-<rOT-i' 



rHCV|fO-<a-iO VOt^OOOvO i-iCSiro-'J-in <0t>>000\0 .-i<Mf*^'<a-to vOt^OOOsO i-HCvarOTj-iO VOt^OOONOT-< jjQ 

^,H,-l,-<r-l rt,-(rt»HT-l ,-<rt,-I^H,-l rti-H^-lr-l ,_|,-l,Hr-l,-t r-»^,-H.-l»-< ,-(^rtrHT-l ^^^^^^ <;C/2 



VOO 

o\cg 
(m';-I 



104 Tests of Mathematical Ability and Their Prognostic Value 





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(2) S1S3X SuipB3^ ajiipujoqx 

(I) SJS3X 3uipB3^ 33lipUJOHX iSSS 

(2) saiBDg aSBnSuBq anqBJX 

(X) S31B3S 33En3uBT anqBJX 

(2) sajisoddQ iBoi^o^ 

(X) sajisoddQ leoiSoq 

(X) suojiBp^ psxij^ 
(2) suiaiqojj opatuqiuv 
(X) sraaiqojjj oi^auimuv 

(X) 3uiU0SB3^ 
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(X) suopiugaQ iBoujatuoaQ 





m 




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s^ 


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<M .-1 <M ,-< .-1 


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CMvoo\oom 

■<l-TMDCSIfC 


evjoj 


<M<N<NCq<N> 


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OnO\vOvO<M 


oo<MO\oom 

CMPOfOi-iCM 


CNirv 

CMCVJ 




.-KM eg Old 


.-» 1-1 1^ t*5 0\ 

(MoqcgrH 


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01<M 


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rHNOfO»-l 


^*CM»-*»H 


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CNJfO 


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oqrowcort 


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fOTflOlOiO 


CM'* 


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t-^OOvOVOf^ 


rcoOiiT!j-iii 


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^i^^-S 


r-l00CMVO<O 


OOVO 




i-lrH(MOJ<M 


ooioo\.-io\ 


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S^ 




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<N CM fO CM CM 


ss 


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Appendix 105 



irtf'VO n«oroiOf»5 T-ii-i^rr>vO t^t^C^•-^^s OOCgOOf^l^ fOOJvOl^vO OO •* C3 lO t>. ^^lrt^^O00»-^ 



•^caoo Tftoo\oovo invoriTj-r^ oot^vooom vooosro'* loovojt^c^ votooo»-iCT\ so ro lo i^ oo o 
voi^t^ iDvoj^oovo i^onoovooo t^r^t>»o\o\ lot^iovor^ oooovovooo t^\£--^t^o\ vovono\ooooo 



vocg 
,-lOOT^ vo'^OxOr^ ocs-uitrstN. rgc\t^'-i"^ ■*t^<r)0\vo ONtrju^Oi-i pr}<M>oCT\vo i^»-hooo\oO'* o'oo 
c\00r^ t^oovoosoo r^oot-xoooo oo»or^t^io oooooot^t^ t^ooooo\t^ t>.ooi^t^oo ooooooooooo 



O\^00 ^.-ivo— iiM 00C\'-<t^O\ CMOOOVOCO OOTj-iDOWO tx0\«-<00 ^OOVOOxO r^t>.oooooo 
.HOJrH C^J<Ni-l(MCNl ^,-|(M,Hi-» ojrHCgi-n-l ,-( CM i-H-H »-l TH rH OJ CVJ N »-« T-l tH .1 <M ^ rH t-H CSJ 1-1 CM 



•<J-(7M-i r>» v£> rH 00 i-H 00>ir)»^vO t^CM'-i-^CM CMCMOOt^ l^ t^ 00 •-< -* *Ot^(MO(M o rg 00 -"i- vo o 
CMi-iCM rt»-H,-i,-icg CM «-! «-!>-« »H .-H i-H CSl tH i-H CM CN •-( CM '"H .-( .-I i-H CM i-H ,-1 ,-1 rH ,-1 (M CO CM •-< CM 1-1 CM 



— <00tM — I -^l- Tj- 00 »o Tj-CMf<000t^ ^»00\0f*5 t^C^fO00t^ t^CMOOOO ONTt-Tfirjio -"S- i-H ro "T<1- o 

C>J.-lt-l CM T-H T-l .-I 1-1 rl CM T-1 r-l rH ^ ^ ,_| fv, ,_| ,_ ,_i ^ ^ rH ,-H CM ^ C^ t-1 .-I ,-1 ,-1 ,-( ^ ,_, ^ ,_,,_,,-) CM 

«^ 0\VO'* CM<*5Tj-\Ot^ Tj-cocOfOOO OOOO'^OO i-H 0^ »-h m to 00r^OvO»-i ^CM>fir>.iO 'i- O rf VO r^ CM 

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g r-^r-. ^r^^^^ _^^^^ ^^^^^ ^^^^_ ^^^^^ _^^^^ ^^^^^^ 

vJ 00-*iO rOi-HOCTi'-t »-n-iro<N100 vO C7\ »n 00 C\ -"l-OOvOVOt^ ui tO t^ O •* CO O (Ml^ t^ NO vO O 0\ t^ 0\ 

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X 

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^ rororo ro ro rH cm CO CSJCM'-hCMCM fO ti CM CM i-i fO CM CM C<l CM CO >-i CM CM CM CVl CM »-i i-l fO CM .-i 

U 

•-) 

CQ 

,*** Oiot^ fOCMtxCMt>» r^iorti-iCN i-IOM-iro^ tOt^cOOO'^ OOOOONt^iO lO'^rooOO t^i-iC<U>»mt^ 

f-H fOC^CM CMf^ fO CMOq rOi-i ro i-i CM t-H CM CM rH CM CM fO i-i CM (M CM CM fC CM 



lOC^O C<lCMO-*»-t roroCvJfOCM i-i r-i CM C<1 i-i tirOTfCMM •* OJ fO m ro t-i VO CM t^ ^4 PO CM CO 't O O 



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•POVO »-iCMO0000 t^OrOfOVO r»5 t^ ^ »-i 00 roO-^ONCM 0\ »*5 to fO 0\ CM lO 0\ OS C<1 \OiOOCMf*iO\ 



vorooo 00ifhOnO\C> >0<50\CMir) 0\>nO0\00 CMOOiotJ-vO vO CvJ i-h C) t>. OsCMONt^CM lOVOCMrOOm 

;o\ 

roC^fO lO lO Tj- ,-( i-H ioO'*VOr-i O\O\V0O00 ^HO'*t^T^ t^ i-t NO 00 »-^ r-i CD tJ- lo t-i CM 0\ CO »n «-! t«» 0»r> 
CMCMCM CM'-'CMCM CM«-<'-'CMCM t-i.-(CM»-i i-ICMrHCMr-i CMCMCMC<) »-<CMCMCMC^ CMC<«OICMtHiH 

o«o 

lOOOO ONfOtOVOO Ti•lF^T^00^^ 0^»ovOcm cm »-• t^ t>> »-< no cm l>» .-i t>» 0\vOiOCM«rt »-• .^ tJ- ro ro oo ovb 
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io6 Tests of Mathematical Ability and Their Prognostic Value 

INSTRUCTIONS TO SUBJECTS 

The directions used in the case of the mathematical tests devised 
for the first time are described in great detail in order that it may be 
possible for any one to repeat their application from the given descrip- 
tion with sufficient similarity of procedure to permit a comparison of 
the results obtained with those recorded in this study. 

The instructions given to the two groups examined were identical. 
The time limit, however, was different in the case of several tests, which 
had been extended, before appHcation to the second group. 

A regular procedure in general technique was followed for all the tests. 
Thus they were invariably presented face down with the warning: "Do 
not turn your paper until you are given the signal 'Go' and stop at 
once, when you hear the signal 'Stop.' " This was followed by the com- 
mand: "Write your name and the date in the upper right-hand corner." 

ALGEBRAIC COMPUTATION: 

"On the other side of the paper in front of you are problems in 
algebra. Work them in the order given. First do 1, then do 2, then 3, 
and so on." 
Time limit: Wadleigh High School, 7 minutes 
Horace Mann School, 12 minutes 
(Sheet I, 4 minutes) 
(Sheet la, 8 minutes) 

MATCHING EQUATIONS AND PROBLEMS: 

"Read the directions. On the other side of the sheet in front of you 
are 12 problems and 12 equations which stand for them. Each problem 
corresponds to one equation, and only one, and each equation stands for 
one problem and only one. You have to match the problems and equa- 
tions. Do not find the answers to the problems. Do not solve the 
equations. Only match the equations and problems. 

"First read problem 1 and state it in the form of an equation, then 
look down the list of equations till you find the right one corresponding 
to problem 1. Then write 1 opposite the equation." 
Time limit: Wadleigh High School (1), 4 minutes 

(1-a), 3 minutes 
Horace Mann School (1), 4 minutes 

(1-a), 10 minutes 

MATCHING iVTH TERMS AND SERIES: 

"Read the directions." 

Write 2» on the board. Ask, "If we substitute for n, 1, what does 
2n equal? If we substitute 2 for n, what does 2n equal? (and so on), 2, 

4, 6, 8, 10 That is a series. How is it formed? How is 4 got from 

2, and 6 from 4, and 8 from 6? What would be the next number after 10? 



Appendix 107 

2, 4, 6, 8, 10, 12, 14, and so on. This series corresponds, or is derived 
from the formula, 2«. The formula is a short way of writing the series 

2, 4, 6, 8 Take the formula 5n-l, what is the series derived from it? 

First let n equal 1, then the formula equals 4, let n equal 2, then the 
formula equals 9, next 14, next 19, next 24, next 29. This series cor- 
responds to 5»-l. What is the term following 29? How is the series 
found? 

"5, 9, 14, 19, 24 corresponds to 5n-l. 

"5»-l is a short way of writing the series 4, 9, 14, 19, 24 

"On the opposite side of this page are 12 formulae and 12 series 
derived from them. You are to pair these correctly, writing in Column 3 
opposite each formula, the number of the series obtained from it. Thus, 
suppose the series obtained from the first formula were the 7th, then 
you would write in Column 3, opposite the first formula the number 7. 
First look at formula I, get from it the series for which it stands. Look 
for the series among the 12 given series and write the number of the 
one selected in Column 3." 

Time limit: Wadleigh High School, 2 minutes. 

Horace Mann School (1), 1^ minutes, 
(la), 2y2 minutes. 

INTERPOLATION TEST: 

Write on the blackboard 2, 4, 6, 8, 10, 12 

Ask: "What is the rule for making this series? 

How is each term got from the one before it? 
How is 4 got from 2? 6 from 4, 8 from 6, etc? 

2, 4,— ,8, 10,—, 14 

What are the missing numbers? 

5, 10, — , 20, 25, — , — 40 

What are the missing steps in this series? 

1, 4,—, 10, — — — , 22 

What are the missing steps in this series?" 
"Write your name and the date. Lay down your pencil." 
"On the other side of this sheet are similar series. They increase in 
difficulty. In the first there are only 2 steps missing, but more and more 
steps are missing, as you go on. You have to fill up each blank space 
with one missing number." 

"Do not turn your paper till I say 'Go' and stop immediately when I 
say 'Stop,' laying down your pencil." 

"You will have — minutes. Your score depends on the number of 
blanks correctly filled." 
Time limit: Wadleigh High School, 2 minutes. 

Horace Mann School (1), 8 minutes, 
(la), 5 minutes. 



io8 Tests of Mathematical Ability and Their Prognostic Value 

MISSING STEPS IN SERIES: 

"Read the directions." 

"Last day you had to find missing numbers in series that were got by 
additions. Always you had to find the number that was added and then 
you were able to fill in the blanks. This time the series are made not 
only by addition, but also by subtraction, multiplication, and division. You 
have to find out in each case which it is. Discover the rule and so 
supply the missing numbers. Look at the illustrations. What is the^rule 
for the first? What is the missing number? What is the rule for the 
second? What is the missing number? the third? the fourth?" 

"Do not turn your paper till I say 'Go,' then turn at once and as fast 
as you can write in the missing numbers. When I say 'Stop,' at once 
lay down your pencil. You will have one minute. Your score depends 
on the number of blanks correctly filled. 

Time limit: Wadleigh High School, 1 minute. 
Horace Mann School, 1 minute. 

INFERENCE WITH SYMBOLS: 

"Write your name and the date." Read the directions. 

"Do not turn over the sheet of paper until told." 

"In algebra you work with symbols. You have already learned to use 
plus (-f ) for add, and minus ( — ) for subtract, and equals (=) for 
equals. Now on this sheet you have new symbols. Look at the illustra- 
tions. The first reads ''A is greater than B, B equals C, therefore, — ? 
A is greater than C The second reads A is greater than B, B is not less 
than C, therefore, — ? ^ is greater than C" 

On the other side there are similar inferences. You are to find the 
conclusions from the "Given Facts" by "filling in" the correct symbols. 
That is, you make an inference from given statements, e.g., if I say A 
is the brother of B, and B is the brother of C, what is the relation be- 
tween A and C, you can tell me that A is the brother of C. The first 
problems are easy: they become more difficult towards the end. Note, 
where none of the symbols give a true conclusion, draw a line. There are 
cases where it is impossible to find any conclusion, i.e., where you cannot 
say whether A is greater than B, less than B, equal to B, not greater 
than B, or not less than B. 

Time limit: Wadleigh High School, 6 minutes. 

Horace Mann School (1 and la), 15 minutes. 
(2 and 2a), 15 minutes. 

GEOMETRY TEST: 

Hand out the reference and problem sheets. Say: "Read the refer- 
ence sheet." Then go over directions carefully with the group. Explain 
the illustration. After they have tried Problem I, explain it. 

N. B. Say: "You may require more than one of the facts to solve the 



Appendix 109 

later problems. Be sure and give them all. You will get a mark for 
each correct reference." 

"Write your name and the date on the back of the Problem sheet." 

(Answer questions) 

Time limit : 30 minutes. 

SUPERPOSITION TEST: 

Prepare three cardboards similar to the three cards shown on the 
instructions side of the Spatial Relations test. Make these cards about 
10 inches on the side. Paint one edge of the card black on both sides of 
the card. Cut the holes as indicated. 

Before giving the test draw on the blackboard the complete drawing 
on the instructions side of the blank. This need not be very accurately 
done. Warn the group that the instructions must be attended to very 
carefully to be understood. Repeat orally the following, while moving 
one of the large cardboards into place on the blackboard drawing. 

"Suppose that the figure with a circle in it is a small card with one 
of its edges painted black and with a hole in one corner. 

"If this card is moved around so that its black edge lies upon the long 
heavy black line, it will fit one of those two figures shown. 

"Decide which it fits and then with your pencil draw a circle where 
the hole would be." 

Give this paragraph verbatim for each of the three tests on the instruc- 
tions side and also make the group try these three tests. 

Then say : "Do the same thing with the other outlines given as quickly 
as you can." 

Time limit: 1 minute. 

This test was applied twice in the case of the Wadleigh High School 
pupils and four times in the case of the Horace Mann group, applications 
being made on two different days. In order to obtain two measures of 
the ability tested the scores for the first two applications were added and 
similarly for the last two applications. 

SYMMETRY TEST (Thurstone Spatial Relations Test) : 

Prepare three cardboards similar to the three cards shown on the 
instructions side of the sheet. Make these cards about ten inches on 
the side. Paint one edge of each card black on both sides of the card. 
Cut the holes as indicated. 

Before giving the test draw on the blackboard the complete drawing on 
the instructions side of the blank. This need not be very accurately 
done. 

Allow two minutes for the group to read the instructions, warning 
them that the instructions must be read very carefully to be understood. 

At the end of this time, while moving one of the large cardboards into 
place on the blackboard, repeat orally this paragraph verbatim for each 
of the three cards on the instruction side. 



no Tests of Mathematical Ability and Their Prognostic Value 

"Imagine that this card is picked up, turned over and placed face down 
with the black edge of the card touching the long heavy black line to 
the right. Imagine the card moved along this black line until its edges 
fit the edges of one or the other of the lozenge shaped outlines. 

"With your pencil, draw a circle in the corner where the hole will be." 

Time limit: Wadleigh High School, 2 minutes. 
Horace Mann School, 2 minutes. 

This test was applied twice to the Wadleigh High School group and 
three times to the Horace Mann pupils. It had been intended to give 
four applications in all to the latter ; but time was not available. The usual 
plan was followed of obtaining two measures of the ability tested. The 
sum of the alternate scores in the third application was added to the 
sum of the scores in the first application and the second application 
respectively, so giving two comparable measures of ability in applying 
the principle of symmetry, 

MATCHING SOLIDS AND SURFACES: 

Give out the reference and the test sheets. 

Allow five minutes for reading the former. Show the actual solids. 
Go over the directions with care, explaining in detail: (1) Matching 
solids and surfaces, (2) method of cutting solids and different surfaces 
obtained by vertical, horizontal, slanting cuts, (3) method of lettering, 
answering any questions upon this. 

Time limit: 30 minutes. 

GEOMETRICAL DEFINITIONS TEST: 

"Read the reference sheet. Do it carefully." 
(Allow two minutes for this) 

"Write your name and the date at the right hand top corner of the 
second sheet." 

"On the reference sheet there are drawings of geometrical figures, and 
definitions of these figures. On the second sheet are different figures and 
you are asked to give complete definitions of these. The reference sheet 
shows the kind of definitions that is wanted. Notice you must give a 
complete and correct definition. The definitions of the new figures will 
be similar, but not exactly the same." 

Time limit : 15 minutes ; usually 90% finish at 12 minutes. 

REASONING TEST : 

Read over the directions with the class and show how the illustrative 
examples are worked. Also say, "The first arguments are very simple 
but grow more and more difficult. You have at first to make only one 
inference, but towards the end you have to make two or three consecutive 
inferences to get the answer and find the required relation." 

Time limit: Wadleigh High School, 10 minutes. 
Horace Mann School, 10^ minutes. 



Appendix iii 

ARITHMETIC PROBLEMS : 
"Do the problems in the order given, first 1, then 2, then 3, and so on." 

Time limit : Wadleigh High School, 10 minutes. 
Horace Mann School, 10 minutes. 

MIXED RELATIONS TEST: 

Write on the board: color — red, name — John, 
page — book handle — 
fire — burns soldiers — 

"The first pair of words express a certain relation. 

"You have to find a fourth word, which along with the third word 
will give the same relation. 

Thus color — red, name — John. That is, Red is a color and John is a 
name. 

"Then ask : Page is to book as handle is to what ? That is : Page 
is a part of a book and handle or blade is part of a knife. 

"Then ask: fire burns soldiers — what? 

"On the other side of the sheet similar relations have to be found. 
You must find a fourth word that is related to the third, as the second 
is related to the first. 

"Your score depends on the number of correct answers." 

Time limit: Wadleigh High School, 1^ minutes. 
Horace Mann School, 3 minutes. 

LOGICAL OPPOSITES TEST: 

"What is the opposite of better?" 

"What is the opposite of friend?" 

"What is the opposite of true?" 

Point out that the answer "untrue" is not as good as "false." 

"On the other side of this sheet there is a list of words. You are to 
write after each one a word that is opposite in meaning to it. You will 
have a minute and a half. Your score depends on the number of right 
opposites written." 

"If you cannot think of the correct opposite within ten seconds, go on 
to the next word." 

Time limit: Wadleigh High School, \y2 minutes. 
Horace Mann School, 5 minutes. 

TRABUE LANGUAGE SCALES (L, M, J, K) : 

Standard directions were followed.^ 

^ See Trabue, M. R., Completion-Test Language Scales, Teachers College, 
Columbia University, Contributions to Education, No. 77. 



112 Tests of Mathematical Ability and Their Prognostic Value 

THORNDIKE READING TESTS : 2 

Wadleigh High School : 

(1) Scale Alpha 2 

(2) Tests I, M, N, N, B, W 
Horace Mann School : 

(1) Scale Alpha 2 

(2) Tests I, M, N, N, R 

"Do exactly what it asks you to do. Answer every question. Bring 
your paper when you have finished so as to get credit for quick work, 
but work very carefully. 

Time limit: 30 minutes. 

2 See Teachers College Record, September, 1914, November, 1915, and 
January, 1916. 



THE PRACTICAL USE OF THE SEXTET OF TESTS FOR 
DIAGNOSING MATHEMATICAL ABILITY 

The object of this sextet of tests is to provide a quick means 
of diagnosing the mathematical inteUigence of pupils in the third 
year of the Junior High School ^ with a view to improving the 
classification of students in high school by eliminating from the 
mathematics classes those unfit for further mathem.atical training 
and selecting those capable of progressing at a more rapid rate 
than the majority. The tests also serve to discover particular 
lines of mathematical weakness. 

The tests recommended are Algebraic Computation, Interpola- 
tion, Geometry, Superposition,* Mixed Relations, and the Trabue 
Language Scales L and J.^ They are designed to measure the 
more important phases of mathematical capacity demanded by 
high school mathematics and in particular the ability to manipulate 
numerical and algebraic symbols, the ability to grasp and handle 
spatial relations, and the ability to deal effectively with words. 
They are of such a nature as to enable an intelligent teacher to 
form an independent estimate of the pupil's mathematical capacity 
and likelihood of success in future mathematical work. They 
measure original ability rather than the effects of training. 

The tests have been applied under differing conditions, how- 
ever, to several hundred persons. The results presented here as 
most valuable for purposes of comparison are those obtained from 
sixty-one pupils in the third year of the Junior High School of 

3 The tests can be given in the seventh and eighth grades. The time 
limits must in these cases be considerably extended and comparative 
standards have not been tabulated. 

*This is a modified form of the Thurstone Spatial Relations test. 

5 See pages 17 to 41 for a description of these tests. The blanks for 
the Superposition test can be obtained from L. L. Thurstone, Carnegie 
Institute of Technology, Pittsburg. The Trabue Language Scales can be 
procured from the Bureau of Publications, Teachers College, Columbia 
University. 

113 



114 Tests of Mathematical Ability and Their Prognostic Value 

the Horace Mann School for Girls. The tests should be ad- 
ministered, under conditions precisely similar to those present in 
their case. They should therefore be given during the second 
half of the school year and the same method of scoring should 
be followed.^ 

The application of the tests demands 72 minutes and if we al- 
low for the preliminary explanations which are necessary, at least 
an hour and a half in time is required for obtaining comparable 
results. The Trabue Language Scales and the Mixed Relations 
test can conveniently be given in the English class hour as a class 
exercise. Two mathematical periods will then complete the appli- 
cation of the four remaining tests. 

The following arrangement is suggested as satisfactory. 



Class Period 
(40 minutes) 



Name of Test 



Time for 
Preliminary 
Explanation 



Time for Test 



Superposition 5 minutes 2 minutes (1 minute 

for each application) 

Algebraic 12 minutes (4 minutes 

Computation for sheet 1 and 8 min- 

utes for sheet la) 

Interpolation 5 minutes 13 minutes (8 minutes 

for sheet 1 and 5 min- 
utes for sheet la) 



II 



Geometry 



10 minutes 30 minutes 



III Mixed Relations 5 minutes 3 minutes 

Trabue Language 

Completion Scales 2 minutes 10 minutes 



The results so obtained should be treated in the following way. 
Each individual's score in each test should be first expressed as a 
deviation from the average mark obtained by the Horace Mann 
group.*^ 



^ For instructions to subjects see the Appendix and for method of 
scoring see the tests, pages 17-41. 

7 The individual scores might also be expressed as deviations from 
their own class average. 



Appendix 115 

The Horace Mann averages were as follows : ® 

Algebraic Computation 20 

Interpolation 92 

Geometry 12 

Superposition 12 

Mixed Relations 21 

Trabue Scales L and J 16 

The general principle underlying the estimation of mathematical 
intelligence is that as many phases as possible of mathematical 
skill and insight should be tested and the results pooled. In 
order to accomplish this it is essential to make the variabilities 
of the various tests equal.® This is done by multiplying the de- 
viations for the Algebraic Computation test by 6, for the Inter- 
polation test by 1, for the Geometry test by 8, for the Superposi- 
tion test by 3, for the Mixed Relations test by 2, and for the Tra- 
bue Scales by 6. These new deviations should then be summed 
algebraically for each individual. The resulting number gives a 
measure of his mathematical capacity. 

In the case of the Horace Mann pupils the composites scores 
so obtained were as follows : 

individual Score Individual Score 

1 34 31 —93 

2 69 32 —29 

3 30 ZZ —98 

4 —89 34 50 

5 —44 35 107 

6 —304 Z6 124 

7 —21 2,7 —91 
8-4 38 —25 
9 46 39 —68 

10 —30 40 —164 

11 —51 41 —85 

12 124 42 —77 

13 32 43 58 

^ The actual averages obtained by the Horace Mann group were : 

Algebraic Computation 20.42 

Interpolation 91.93 

Geometry 11.96 

Superposition 12.03 

Mixed Relations 20.59 

Trabue Scales L and J 16.42 

To simplify calculation the nearest integer is recommended for use, 
however, being sufficiently accurate. 

9 See page 66. 



ii6 Tests of Mathematical Ability and Their Prognostic Value 



Individual 


^'cor*? 


Individual 


Score 


14 


—179 


44 


—20 


15 


—72 


45 


-67 


16 


100 


46 


-95 


17 


121 


47 


—121 


18 


337 


48 


176 


19 


95 


49 


289 


20 


216 


50 


—171 


21 


12 


51 


—93 


22 


—3 


52 


182 


23 


218 


53 


-57 


24 


103 


54 





25 


83 


55 


—163 


26 


38 


56 


—70 


27 


—34 


57 


31 


28 


—90 


58 


98 


29 


164 


59 


129 


30 


—29 


60 


—169 






61 


—143 



These scores are represented graphically in the accompanying 
figure. 



+ 

4- 

+ + 

+ '+ + 

+ + + 4- + + 

-f + + 4-+ 4- + + 

+ + + + + + ■+ + 

4 +• + + + + + +-f+ + + 

-}- 4- + 4- + 4-4- 4--I-4-4-4-4-4-4-4-4- 



-400 -300 -200 -iOO MOO +Z00 +300 *400 

As tentative standards we suggest (1) where a pupil's score 
is greater than 150, he has capacity to progress at a more rapid 
rate than the ordinary high school student; (2) where a pupil's 
score is less than — 150, he shows incapacity to progress in mathe- 
matics at the rate of the ordinary high school student and, other 
things being equal, should be released from further training in the 
subject. 

These standards are tentative, but they err on the safe side.^° 

10 The reliability of the composite score for the sextet of tests has been 
estimated and is satisfactorily high. The results of two independent 
applications to the Horace Mann group gave a reliability coefficient of 



Appendix 



117 






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Ii8 Tests of Mathematical Ability and Their Prognostic Value 

When any doubt is felt with regard to the ability of a pupil the 
tests should be re-applied in duplicate. Duplicates of the tests 
exist and can be supplied by the writer. More reliable standards 
will be established by application of the tests to larger numbers 
of children. Tests will be supplied at cost to those who will fur- 
nish results to the writer. 

For the discovery of particular lines of mathematical weakness 
in the individual pupils use can be made of the graph to locate 
special disabilities, which is given on page 117. It provides a 
record of the relations of a pupil's abilities in mathematics to the 
abilities of others. 

Each test is represented in the graph by a horizontal line. The 
scales are so drawn that the average marks for the six tests lie 
on a straight line. In the case of the Interpolation test all the 
scores are not directly indicated, but they can be roughly placed, 
when the individual curve is drawn. The interpretation of 
curves is simple. For example, in the graph, pupil A is seen 
to be above the average in all the tests, excelling especially 
in the test of intuitive grasp of spatial relations. Pupil C, on 
the other hand, is below the average in all save the Superposition 
test, failing conspicuously in ability to grasp spatial and abstract 
relations. Pupil B is above the average to a slight extent save 
in the Superposition test and the Trabue Language Scales, al- 
though he is not especially weak in the latter. 

The graph to locate special disabilities can thus be profitably 
used as a check upon the opinions arrived at by the mathematics 
teacher as to the pupil's lines of strength and weakness. 

.92 ±: .01. Here r is seventy times as large as its Probable Error. Its 
reliability, therefore, is very high. 

In the case of the Horace Mann group the equation for estimating an 
individual's score in school marks from his score in the tests is the 
regression equation: x=.Q23y or, v/here the score in the tests is left in 
terms of deviation from the class average and x is replaced by the abso- 
lute value of the variable, namely, X — 12.44, it becomes: X=:l2.26-\-.023y. 
The standard error in using this equation is 1.01 (see Yule, G. Udny., An 
Introduction to the Theory of Statistics, London, 1916: 177). 



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